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<title>Probability space</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Probability space</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical concept. For the novel, see <a href="Probability_Space_(novel)" class="mw-redirect" title="Probability Space (novel)">Probability Space (novel)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<ul><li><a href="Probability" title="Probability">Probability</a>

<ul><li><a href="Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>

<li><a href="Determinism" title="Determinism">Determinism</a>

<ul><li><a href="Deterministic_system" title="Deterministic system">System</a></li></ul></li>

<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>

<li><a href="Randomness" title="Randomness">Randomness</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul>

<li><a href="Sample_space" title="Sample space">Sample space</a></li>

<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event</a>

<ul><li><a href="Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>

<li><a href="Elementary_event" title="Elementary event">Elementary event</a></li>

<li><a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>

<li><a href="Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>

<li><a href="Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>

<ul><li><a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a>

<ul><li><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>

<li><a href="Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>

<li><a href="Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>

<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>

<li><a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>

<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>

<li><a href="Probability_measure" title="Probability measure">Probability measure</a></li>

<li><a href="Random_variable" title="Random variable">Random variable</a>

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>

<li><a href="Expected_value" title="Expected value">Expected value</a></li>

<li><a href="Variance" title="Variance">Variance</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Realization_(probability)" title="Realization (probability)">Observed value</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a></li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Complementary_event" title="Complementary event">Complementary event</a></li>

<li><a href="Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>

<li><a href="Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>

<li><a href="Conditional_independence" title="Conditional independence">Conditional independence</a></li>

<li><a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>

<li><a href="Boole's_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li>

<li><a href="Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>

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<p>In <a href="Probability_theory" title="Probability theory">probability theory</a>, a <b>probability space</b> or a <b>probability triple</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

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</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> is a <a href="Space_(mathematics)" title="Space (mathematics)">mathematical construct</a> that provides a formal model of a <a href="Randomness" title="Randomness">random</a> process or "experiment". For example, one can define a probability space which models the throwing of a <a href="Dice" title="Dice">die</a>.

</p><p>A probability space consists of three elements:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>A <i><a href="Sample_space" title="Sample space">sample space</a></i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, which is the set of all possible <a href="Outcome_(probability)" title="Outcome (probability)">outcomes</a> of a random process under consideration.</li>

<li>An <b>event space</b>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>, which is a set of <a href="Event_(probability_theory)" title="Event (probability theory)">events</a>, where an event is a subset of outcomes in the sample space.</li>

<li>A <i><a href="Probability_measure" title="Probability measure">probability function</a></i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, which assigns, to each event in the event space, a <a href="Probability" title="Probability">probability</a>, which is a number between 0 and 1 (inclusive).</li></ol>

<p>In order to provide a model of probability, these elements must satisfy <a href="Probability_axioms" title="Probability axioms">probability axioms</a>.

</p><p>In the example of the throw of a standard die,

</p>

<ol><li>The sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is typically the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4,5,6\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4,5,6\}}</annotation>

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</math></span><img src="./bc427c339db79f243cb79154253ff8151a31c23e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.469ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4,5,6\}}" loading="lazy"></span> where each element in the set is a label which represents the outcome of the die landing on that label. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> represents the outcome that the die lands on 1.</li>

<li>The event space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> could be the <a href="Power_set" title="Power set">set of all subsets</a> of the sample space, which would then contain simple events such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{5\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{5\}}</annotation>

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</math></span><img src="./3c0761b1f7e017a291a4d7fe4beac96d877d9441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{5\}}" loading="lazy"></span> ("the die lands on 5"), as well as complex events such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,4,6\}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{2,4,6\}}</annotation>

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</math></span><img src="./dc24923d36d36829caf41d4e801eeb5409bb0766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,4,6\}}" loading="lazy"></span> ("the die lands on an even number").</li>

<li>The probability function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> would then map each event to the number of outcomes in that event divided by 6 – so for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{5\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{5\}}</annotation>

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</math></span><img src="./3c0761b1f7e017a291a4d7fe4beac96d877d9441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{5\}}" loading="lazy"></span> would be mapped to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/6}">

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<annotation encoding="application/x-tex">{\displaystyle 1/6}</annotation>

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</math></span><img src="./365dd517702686649e6644a66665efd1a32be3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/6}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,4,6\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{2,4,6\}}</annotation>

</semantics>

</math></span><img src="./dc24923d36d36829caf41d4e801eeb5409bb0766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{2,4,6\}}" loading="lazy"></span> would be mapped to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3/6=1/2}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3/6=1/2}</annotation>

</semantics>

</math></span><img src="./bdaba543df96ede625625b58a25932ec2b1f22ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.073ex; height:2.843ex;" alt="{\displaystyle 3/6=1/2}" loading="lazy"></span>.</li></ol>

<p>When an experiment is conducted, it results in exactly one outcome <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> from the sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. All the events in the event space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> that contain the selected outcome <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> are said to "have occurred". The probability function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> must be so defined that if the experiment were repeated arbitrarily many times, the number of occurrences of each event as a fraction of the total number of experiments, will most likely tend towards the probability assigned to that event.

</p><p>The Soviet mathematician <a href="Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> introduced the notion of a probability space and the <a href="Axioms_of_probability" class="mw-redirect" title="Axioms of probability">axioms of probability</a> in the 1930s. In modern probability theory, there are alternative approaches for axiomatization, such as the <a href="Algebra_of_random_variables" title="Algebra of random variables">algebra of random variables</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>

<p>A probability space is a mathematical triplet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> that presents a <a href="Mathematical_model" title="Mathematical model">model</a> for a particular class of real-world situations. As with other models, its author ultimately defines which elements <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> will contain.

</p>

<ul><li>The <a href="Sample_space" title="Sample space">sample space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is the set of all possible outcomes. An <a href="Outcome_(probability)" title="Outcome (probability)">outcome</a> is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do. This leads to different choices of sample space.</li>

<li>The <a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is a collection of all the <a href="Event_(probability_theory)" title="Event (probability theory)">events</a> we would like to consider. This collection may or may not include each of the <a href="Elementary_event" title="Elementary event">elementary</a> events. Here, an "event" is a set of zero or more outcomes; that is, a <a href="Subset" title="Subset">subset</a> of the sample space. An event is considered to have "happened" during an experiment when the outcome of the latter is an element of the event. Since the same outcome may be a member of many events, it is possible for many events to have happened given a single outcome. For example, when the trial consists of throwing two dice, the set of all outcomes with a sum of 7 <a href="Pip_(counting)" title="Pip (counting)">pips</a> may constitute an event, whereas outcomes with an odd number of pips may constitute another event. If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events, "7 pips" and "odd number of pips", are said to have happened.</li>

<li>The <a href="Probability_measure" title="Probability measure">probability measure</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is a <a href="Set_function" title="Set function">set function</a> returning an event's <a href="Probability" title="Probability">probability</a>. A probability is a real number between zero (impossible events have probability zero, though probability-zero events are not necessarily impossible) and one (the event happens <a href="Almost_surely" title="Almost surely">almost surely</a>, with almost total certainty). Thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:{\mathcal {F}}\to [0,1].}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P:{\mathcal {F}}\to [0,1].}</annotation>

</semantics>

</math></span><img src="./d0f32747edc817e58b7a3344b7a77b6f38b0aa0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.523ex; height:2.843ex;" alt="{\displaystyle P:{\mathcal {F}}\to [0,1].}" loading="lazy"></span> The probability measure function must satisfy two simple requirements: First, the probability of a <a href="Countable_set" title="Countable set">countable</a> union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. For example, the probability of the union of the mutually exclusive events <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Head}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Head}}}</annotation>

</semantics>

</math></span><img src="./ff160b97f8e9dfbc507b2bf154eeafa3a2587292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.23ex; height:2.176ex;" alt="{\displaystyle {\text{Head}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tail}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Tail}}}</annotation>

</semantics>

</math></span><img src="./2eff71a39ddcb493aa3e76e4086f68a3601ea096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\displaystyle {\text{Tail}}}" loading="lazy"></span> in the random experiment of one coin toss, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{Head}}\cup {\text{Tail}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P({\text{Head}}\cup {\text{Tail}})}</annotation>

</semantics>

</math></span><img src="./c679286482da8cdf168f2d5584cbc0d3cde92edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.502ex; height:2.843ex;" alt="{\displaystyle P({\text{Head}}\cup {\text{Tail}})}" loading="lazy"></span>, is the sum of probability for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Head}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Head}}}</annotation>

</semantics>

</math></span><img src="./ff160b97f8e9dfbc507b2bf154eeafa3a2587292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.23ex; height:2.176ex;" alt="{\displaystyle {\text{Head}}}" loading="lazy"></span> and the probability for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tail}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Tail}}}</annotation>

</semantics>

</math></span><img src="./2eff71a39ddcb493aa3e76e4086f68a3601ea096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\displaystyle {\text{Tail}}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\text{Head}})+P({\text{Tail}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P({\text{Head}})+P({\text{Tail}})}</annotation>

</semantics>

</math></span><img src="./c293a2cf82bcf498ede78a2b82c72d67a55a08cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.315ex; height:2.843ex;" alt="{\displaystyle P({\text{Head}})+P({\text{Tail}})}" loading="lazy"></span>. Second, the probability of the sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> must be equal to 1 (which accounts for the fact that, given an execution of the model, some outcome must occur). In the previous example the probability of the set of outcomes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{{\text{Head}},{\text{Tail}}\})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{{\text{Head}},{\text{Tail}}\})}</annotation>

</semantics>

</math></span><img src="./9ed0767f6b683e53d37389ae1988a793dd79f248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.278ex; height:2.843ex;" alt="{\displaystyle P(\{{\text{Head}},{\text{Tail}}\})}" loading="lazy"></span> must be equal to one, because it is entirely certain that the outcome will be either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Head}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Head}}}</annotation>

</semantics>

</math></span><img src="./ff160b97f8e9dfbc507b2bf154eeafa3a2587292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.23ex; height:2.176ex;" alt="{\displaystyle {\text{Head}}}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tail}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Tail}}}</annotation>

</semantics>

</math></span><img src="./2eff71a39ddcb493aa3e76e4086f68a3601ea096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\displaystyle {\text{Tail}}}" loading="lazy"></span> (the model neglects any other possibility) in a single coin toss.</li></ul>

<p>Not every subset of the sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be <a href="Non-measurable_set" title="Non-measurable set">"measured"</a>. This is not so obvious in a case like a coin toss. In a different example, one could consider javelin throw lengths, where the events typically are intervals like "between 60 and 65 meters" and unions of such intervals, but not sets like the "irrational numbers between 60 and 65 meters".

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>In short, a probability space is a <a href="Measure_space" title="Measure space">measure space</a> such that the measure of the whole space is equal to one.

</p><p>The expanded definition is the following: a probability space is a triple <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> consisting of:

</p>

<ul><li>the <a href="Sample_space" title="Sample space">sample space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> – an arbitrary <a href="Non-empty_set" class="mw-redirect" title="Non-empty set">non-empty set</a>,</li>

<li>the <a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./2b9e723df202ac23d9b6f6085f41d3abb1e3318f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.606ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}" loading="lazy"></span> (also called σ-field) – a set of subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, called <a href="Event_(probability_theory)" title="Event (probability theory)">events</a>, such that:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> contains the sample space: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./e58d6c9cfaccd8b03c9cb1acffb3857023613120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle \Omega \in {\mathcal {F}}}" loading="lazy"></span>,</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is closed under <a href="Complement_(set_theory)" title="Complement (set theory)">complements</a>: if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy"></span>, then also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega \setminus A)\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega \setminus A)\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./9fe0e23b22ff71c8ef231dfd1b5f1825f0f7cce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.192ex; height:2.843ex;" alt="{\displaystyle (\Omega \setminus A)\in {\mathcal {F}}}" loading="lazy"></span>,</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is closed under <a href="Countable_set" title="Countable set">countable</a> <a href="Union_(set_theory)" title="Union (set theory)">unions</a>: if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}\in {\mathcal {F}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./32d4c92ff217e6345488aed803bcb68c34d50240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.509ex;" alt="{\displaystyle A_{i}\in {\mathcal {F}}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,\dots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots }</annotation>

</semantics>

</math></span><img src="./909e32b58e157a04140957ff66df338a836f7f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.017ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots }" loading="lazy"></span>, then also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\bigcup _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (\bigcup _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./7efb6defa3c19c0acf3720a300825eb1e339b31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.343ex; height:3.176ex;" alt="{\textstyle (\bigcup _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}" loading="lazy"></span>

<ul><li>The corollary from the previous two properties and <a href="De_Morgan's_law" class="mw-redirect" title="De Morgan's law">De Morgan's law</a> is that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is also closed under countable <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersections</a>: if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}\in {\mathcal {F}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./32d4c92ff217e6345488aed803bcb68c34d50240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.509ex;" alt="{\displaystyle A_{i}\in {\mathcal {F}}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,\dots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots }</annotation>

</semantics>

</math></span><img src="./909e32b58e157a04140957ff66df338a836f7f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.017ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots }" loading="lazy"></span>, then also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\bigcap _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (\bigcap _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./4c1445e79a877403b6987ccab5d5d093a8f80247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.343ex; height:3.176ex;" alt="{\textstyle (\bigcap _{i=1}^{\infty }A_{i})\in {\mathcal {F}}}" loading="lazy"></span></li></ul></li></ul></li>

<li>the <a href="Probability_measure" title="Probability measure">probability measure</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:{\mathcal {F}}\to [0,1]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P:{\mathcal {F}}\to [0,1]}</annotation>

</semantics>

</math></span><img src="./17838c21119a9afcb78754109363101442ebf56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.876ex; height:2.843ex;" alt="{\displaystyle P:{\mathcal {F}}\to [0,1]}" loading="lazy"></span> – a function on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> such that:

<ul><li><i>P</i> is <a href="Countably_additive" class="mw-redirect" title="Countably additive">countably additive</a> (also called σ-additive): if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./66d056a48c70562c42c27c17b4b1f2eba91c3e0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.793ex; height:3.009ex;" alt="{\displaystyle \{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}}" loading="lazy"></span> is a countable collection of pairwise <a href="Disjoint_sets" title="Disjoint sets">disjoint sets</a>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i}),}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i}),}</annotation>

</semantics>

</math></span><img src="./99a0ba9ad27f5f4d1665b6a677a3833afdb24cb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.905ex; height:3.176ex;" alt="{\textstyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i}),}" loading="lazy"></span></li>

<li>the measure of the entire sample space is equal to one: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\Omega )=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\Omega )=1}</annotation>

</semantics>

</math></span><img src="./a66a50d3879f50fc0efb4fbd61cdc832b823640b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.494ex; height:2.843ex;" alt="{\displaystyle P(\Omega )=1}" loading="lazy"></span>.</li></ul></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Discrete_case">Discrete case</h2></div>

<p><a href="Discrete_probability_theory" class="mw-redirect" title="Discrete probability theory">Discrete probability theory</a> needs only <a href="Countable_set" title="Countable set">at most countable</a> sample spaces <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>. Probabilities can be ascribed to points of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> by the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:\Omega \to [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:\Omega \to [0,1]}</annotation>

</semantics>

</math></span><img src="./13fceaee29eff4e9294472f749d4b0c31d5c8438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.141ex; height:2.843ex;" alt="{\displaystyle p:\Omega \to [0,1]}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{\omega \in \Omega }p(\omega )=1}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{\omega \in \Omega }p(\omega )=1}</annotation>

</semantics>

</math></span><img src="./5447f81b1a600f707674b8b46d71944efd106968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.064ex; height:3.009ex;" alt="{\textstyle \sum _{\omega \in \Omega }p(\omega )=1}" loading="lazy"></span>. All subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> can be treated as events (thus, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> is the <a href="Power_set" title="Power set">power set</a>). The probability measure takes the simple form

</p>

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/* end https://en.wikipedia.org/ */

</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)=\sum _{\omega \in A}p(\omega )\quad {\text{for all }}A\subseteq \Omega .}">

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(A)=\sum _{\omega \in A}p(\omega )\quad {\text{for all }}A\subseteq \Omega .}</annotation>

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</math></span></span></td> <td></td> <td class="nowrap"><span id="math_⁎" class="reference nourlexpansion" style="font-weight:bold;">⁎</span></td></tr></tbody></table>

<p>The greatest σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> describes the complete information. In general, a σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./2b9e723df202ac23d9b6f6085f41d3abb1e3318f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.606ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }}" loading="lazy"></span> corresponds to a finite or countable <a href="Partition_of_a_set" title="Partition of a set">partition</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =B_{1}\cup B_{2}\cup \dots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =B_{1}\cup B_{2}\cup \dots }</annotation>

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</math></span><img src="./97e093414c18484e0e18405b21623a7c689a52ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.301ex; height:2.509ex;" alt="{\displaystyle \Omega =B_{1}\cup B_{2}\cup \dots }" loading="lazy"></span>, the general form of an event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy"></span> being <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B_{k_{1}}\cup B_{k_{2}}\cup \dots }">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=B_{k_{1}}\cup B_{k_{2}}\cup \dots }</annotation>

</semantics>

</math></span><img src="./fb10ef963446c4d4de1b19f6481f3ba78556f055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.099ex; height:2.843ex;" alt="{\displaystyle A=B_{k_{1}}\cup B_{k_{2}}\cup \dots }" loading="lazy"></span>. See also the examples.

</p><p>The case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\omega )=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\omega )=0}</annotation>

</semantics>

</math></span><img src="./de63695e19130daef13240552e8c3f00dee719d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.775ex; height:2.843ex;" alt="{\displaystyle p(\omega )=0}" loading="lazy"></span> is permitted by the definition, but rarely used, since such <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> can safely be excluded from the sample space.

</p>

<div class="mw-heading mw-heading2"><h2 id="General_case">General case</h2></div>

<p>If <span class="texhtml">Ω</span> is <a href="Uncountable_set" title="Uncountable set">uncountable</a>, still, it may happen that <span class="texhtml"><i>P</i>(<i>ω</i>) ≠ 0</span> for some <span class="texhtml"><i>ω</i></span>; such <span class="texhtml"><i>ω</i></span> are called <a href="Atom_(measure_theory)" title="Atom (measure theory)">atoms</a>. They are an at most countable (maybe <a href="Empty_set" title="Empty set">empty</a>) set, whose probability is the sum of probabilities of all atoms. If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case. Otherwise, if the sum of probabilities of all atoms is between 0 and 1, then the probability space decomposes into a discrete (atomic) part (maybe empty) and a <a href="Atom_(measure_theory)" title="Atom (measure theory)">non-atomic</a> part.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-atomic_case">Non-atomic case</h2></div>

<p>If <span class="texhtml"><i>P</i>(<i>ω</i>) = 0</span> for all <span class="texhtml"><i>ω</i> ∈ Ω</span> (in this case, Ω must be uncountable, because otherwise <span class="texhtml">P(Ω) = 1</span> could not be satisfied), then equation (<b><a href="#math_⁎">⁎</a></b>) fails: the probability of a set is not necessarily the sum over the probabilities of its elements, as summation is only defined for countable numbers of elements. This makes the probability space theory much more technical. A formulation stronger than summation, <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> is applicable. Initially the probabilities are ascribed to some "generator" sets (see the examples). Then a limiting procedure allows assigning probabilities to sets that are limits of sequences of generator sets, or limits of limits, and so on. All these sets are the σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>. For technical details see <a href="Carath%C3%A9odory's_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a>. Sets belonging to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> are called <a href="Measurable" class="mw-redirect" title="Measurable">measurable</a>. In general they are much more complicated than generator sets, but much better than <a href="Non-measurable_set" title="Non-measurable set">non-measurable sets</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Complete_probability_space">Complete probability space</h2></div>

<p>A probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,\;{\mathcal {F}},\;P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,\;{\mathcal {F}},\;P)}</annotation>

</semantics>

</math></span><img src="./f7998d88d53b50a8bcfc418a9d3d0e454e4464eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.518ex; height:2.843ex;" alt="{\displaystyle (\Omega ,\;{\mathcal {F}},\;P)}" loading="lazy"></span> is said to be a complete probability space if for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./6335bdc79abc778e50a43797e0f26f3c121be3d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.531ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {F}}}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(B)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(B)=0}</annotation>

</semantics>

</math></span><img src="./973532626488736dc14f2ab2add7c09c7b2132ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.58ex; height:2.843ex;" alt="{\displaystyle P(B)=0}" loading="lazy"></span> and all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;\subset \;B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\;\subset \;B}</annotation>

</semantics>

</math></span><img src="./b0de6c446e784a7749072f7ef3271b220a89b503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.896ex; height:2.176ex;" alt="{\displaystyle A\;\subset \;B}" loading="lazy"></span> one has <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./8cdf27fd56b3c06d1ddf9ad1efd7cee0a81cb2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.51ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {F}}}" loading="lazy"></span>. Often, the study of probability spaces is restricted to complete probability spaces.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Discrete_examples">Discrete examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4></div>

<p>If the experiment consists of just one flip of a <a href="Fair_coin" title="Fair coin">fair coin</a>, then the outcome is either heads or tails: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\{{\text{H}},{\text{T}}\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{{\text{H}},{\text{T}}\}}</annotation>

</semantics>

</math></span><img src="./e0c3ec8fb49ec9bdaef0a33567b5d2629ef65868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.557ex; height:2.843ex;" alt="{\displaystyle \Omega =\{{\text{H}},{\text{T}}\}}" loading="lazy"></span>. The σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> contains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}=4}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{2}=4}</annotation>

</semantics>

</math></span><img src="./544566e539538b9b5d5f9106712550b2c80da685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 2^{2}=4}" loading="lazy"></span> events, namely: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{H}}\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\text{H}}\}}</annotation>

</semantics>

</math></span><img src="./b16dc960e35800d2b188ae79dbd687c8313600ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle \{{\text{H}}\}}" loading="lazy"></span> ("heads"), <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{T}}\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\text{T}}\}}</annotation>

</semantics>

</math></span><img src="./2767943f07a281e3012d7695f9f245369eee301e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.003ex; height:2.843ex;" alt="{\displaystyle \{{\text{T}}\}}" loading="lazy"></span> ("tails"), <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation>

</semantics>

</math></span><img src="./3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}" loading="lazy"></span> ("neither heads nor tails"), and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\text{H}},{\text{T}}\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\text{H}},{\text{T}}\}}</annotation>

</semantics>

</math></span><img src="./79c5302336f7f76441b006b46d8bf9358491087a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.78ex; height:2.843ex;" alt="{\displaystyle \{{\text{H}},{\text{T}}\}}" loading="lazy"></span> ("either heads or tails"); in other words, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=\{\{\},\{{\text{H}}\},\{{\text{T}}\},\{{\text{H}},{\text{T}}\}\}}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\{\{\},\{{\text{H}}\},\{{\text{T}}\},\{{\text{H}},{\text{T}}\}\}}</annotation>

</semantics>

</math></span><img src="./97203cb99cec0d6d1b3e827c173cb4950d7c457d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.628ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}=\{\{\},\{{\text{H}}\},\{{\text{T}}\},\{{\text{H}},{\text{T}}\}\}}" loading="lazy"></span>. There is a fifty percent chance of tossing heads and fifty percent for tails, so the probability measure in this example is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{\})=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{\})=0}</annotation>

</semantics>

</math></span><img src="./2a9d958e285f771a9f6f49a84b650998e37b1e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.141ex; height:2.843ex;" alt="{\displaystyle P(\{\})=0}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{{\text{H}}\})=0.5}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{{\text{H}}\})=0.5}</annotation>

</semantics>

</math></span><img src="./5c774c6600ff0d6f7e45d5d16ebdb45da624d7dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.693ex; height:2.843ex;" alt="{\displaystyle P(\{{\text{H}}\})=0.5}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{{\text{T}}\})=0.5}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{{\text{T}}\})=0.5}</annotation>

</semantics>

</math></span><img src="./6afdf04efb32b0643417b2213ed14853fbcd9605.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.628ex; height:2.843ex;" alt="{\displaystyle P(\{{\text{T}}\})=0.5}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{{\text{H}},{\text{T}}\})=1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{{\text{H}},{\text{T}}\})=1}</annotation>

</semantics>

</math></span><img src="./f4dc8370ba641d1d3c6ec25c661e8df5c80ab137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.596ex; height:2.843ex;" alt="{\displaystyle P(\{{\text{H}},{\text{T}}\})=1}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4></div>

<p>The fair coin is tossed three times. There are 8 possible outcomes: <span class="texhtml">Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}</span> (here "HTH" for example means that first time the coin landed heads, the second time tails, and the last time heads again). The complete information is described by the σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> of <span class="texhtml">2<sup>8</sup> = 256</span> events, where each of the events is a subset of Ω.

</p><p>Alice knows the outcome of the second toss only. Thus her incomplete information is described by the partition <span class="texhtml">Ω = <i>A</i><sub>1</sub> ⊔ <i>A</i><sub>2</sub> = {HHH, HHT, THH, THT} ⊔ {HTH, HTT, TTH, TTT}</span>, where ⊔ is the <i><a href="Disjoint_union" title="Disjoint union">disjoint union</a></i>, and the corresponding σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Alice}}=\{\{\},A_{1},A_{2},\Omega \}}">

<msub>

</mrow>

<mtext>Alice</mtext>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Alice}}=\{\{\},A_{1},A_{2},\Omega \}}</annotation>

</semantics>

</math></span><img src="./1af85eef9eb3bd14ad09aaac58efdac85b7614be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.634ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Alice}}=\{\{\},A_{1},A_{2},\Omega \}}" loading="lazy"></span>. Bryan knows only the total number of tails. His partition contains four parts: <span class="texhtml">Ω = <i>B</i><sub>0</sub> ⊔ <i>B</i><sub>1</sub> ⊔ <i>B</i><sub>2</sub> ⊔ <i>B</i><sub>3</sub> = {HHH} ⊔ {HHT, HTH, THH} ⊔ {TTH, THT, HTT} ⊔ {TTT}</span>; accordingly, his σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Bryan}}}">

<msub>

</mrow>

<mtext>Bryan</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Bryan}}}</annotation>

</semantics>

</math></span><img src="./17c9416db38737870211428ace7d2840515398b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.315ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Bryan}}}" loading="lazy"></span> contains 2<sup>4</sup> = 16 events.

</p><p>The two σ-algebras are <a href="Comparability" title="Comparability">incomparable</a>: neither <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Alice}}\subseteq {\mathcal {F}}_{\text{Bryan}}}">

<msub>

</mrow>

<mtext>Alice</mtext>

</mrow>

</msub>

<msub>

</mrow>

<mtext>Bryan</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Alice}}\subseteq {\mathcal {F}}_{\text{Bryan}}}</annotation>

</semantics>

</math></span><img src="./fd90be6e41cc85a373ee6cfa032f7c1ba9264281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.925ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Alice}}\subseteq {\mathcal {F}}_{\text{Bryan}}}" loading="lazy"></span> nor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Bryan}}\subseteq {\mathcal {F}}_{\text{Alice}}}">

<msub>

</mrow>

<mtext>Bryan</mtext>

</mrow>

</msub>

<msub>

</mrow>

<mtext>Alice</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Bryan}}\subseteq {\mathcal {F}}_{\text{Alice}}}</annotation>

</semantics>

</math></span><img src="./49b2aa2577e111732fc736ebc3d0786b95305c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.925ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Bryan}}\subseteq {\mathcal {F}}_{\text{Alice}}}" loading="lazy"></span>; both are sub-σ-algebras of 2<sup>Ω</sup>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Example_3">Example 3</h4></div>

<p>If 100 voters are to be drawn randomly from among all voters in California and asked whom they will vote for governor, then the set of all <a href="Sequence" title="Sequence">sequences</a> of 100 Californian voters would be the sample space Ω. We assume that <a href="Simple_random_sample" title="Simple random sample">sampling without replacement</a> is used: only sequences of 100 <i>different</i> voters are allowed. For simplicity an ordered sample is considered, that is a sequence (Alice, Bryan) is different from (Bryan, Alice). We also take for granted that each potential voter knows exactly his/her future choice, that is he/she does not choose randomly.

</p><p>Alice knows only whether or not <a href="Arnold_Schwarzenegger" title="Arnold Schwarzenegger">Arnold Schwarzenegger</a> has received at least 60 votes. Her incomplete information is described by the σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Alice}}}">

<msub>

</mrow>

<mtext>Alice</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Alice}}}</annotation>

</semantics>

</math></span><img src="./8c63007cb67cab3e36c301ad75db219346591468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.511ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Alice}}}" loading="lazy"></span> that contains: (1) the set of all sequences in Ω where at least 60 people vote for Schwarzenegger; (2) the set of all sequences where fewer than 60 vote for Schwarzenegger; (3) the whole sample space Ω; and (4) the empty set ∅.

</p><p>Bryan knows the exact number of voters who are going to vote for Schwarzenegger. His incomplete information is described by the corresponding partition <span class="texhtml">Ω = <i>B</i><sub>0</sub> ⊔ <i>B</i><sub>1</sub> ⊔ ⋯ ⊔ <i>B</i><sub>100</sub></span> and the σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Bryan}}}">

<msub>

</mrow>

<mtext>Bryan</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Bryan}}}</annotation>

</semantics>

</math></span><img src="./17c9416db38737870211428ace7d2840515398b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.315ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Bryan}}}" loading="lazy"></span> consists of 2<sup>101</sup> events.

</p><p>In this case, Alice's σ-algebra is a subset of Bryan's: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{\text{Alice}}\subset {\mathcal {F}}_{\text{Bryan}}}">

<msub>

</mrow>

<mtext>Alice</mtext>

</mrow>

</msub>

<msub>

</mrow>

<mtext>Bryan</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{\text{Alice}}\subset {\mathcal {F}}_{\text{Bryan}}}</annotation>

</semantics>

</math></span><img src="./9645f5d211f0ac123401d7cd2fdd500fbf070a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.925ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{\text{Alice}}\subset {\mathcal {F}}_{\text{Bryan}}}" loading="lazy"></span>. Bryan's σ-algebra is in turn a subset of the much larger "complete information" σ-algebra 2<sup>Ω</sup> consisting of <span class="texhtml">2<sup><i>n</i>(<i>n</i>−1)⋯(<i>n</i>−99)</sup></span> events, where <i>n</i> is the number of all potential voters in California.

</p>

<div class="mw-heading mw-heading3"><h3 id="Non-atomic_examples">Non-atomic examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Example_4">Example 4</h4></div>

<p>A number between 0 and 1 is chosen at random, uniformly. Here Ω = [0,1], <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is the σ-algebra of <a href="Borel_set" title="Borel set">Borel sets</a> on Ω, and <i>P</i> is the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on [0,1].

</p><p>In this case, the open intervals of the form <span class="texhtml">(<i>a</i>,<i>b</i>)</span>, where <span class="texhtml">0 < <i>a</i> < <i>b</i> < 1</span>, could be taken as the generator sets. Each such set can be ascribed the probability of <span class="texhtml"><i>P</i>((<i>a</i>,<i>b</i>)) = (<i>b</i> − <i>a</i>)</span>, which generates the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on [0,1], and the <a href="Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a> on Ω.

</p>

<div class="mw-heading mw-heading4"><h4 id="Example_5">Example 5</h4></div>

<p>A fair coin is tossed endlessly. Here one can take Ω = {0,1}<sup>∞</sup>, the set of all infinite sequences of numbers 0 and 1. <a href="Cylinder_set" title="Cylinder set">Cylinder sets</a> <span class="texhtml">{(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ...) ∈ Ω : <i>x</i><sub>1</sub> = <i>a</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub> = <i>a</i><sub><i>n</i></sub>}</span> may be used as the generator sets. Each such set describes an event in which the first <i>n</i> tosses have resulted in a fixed sequence <span class="texhtml">(<i>a</i><sub>1</sub>, ..., <i>a</i><sub><i>n</i></sub>)</span>, and the rest of the sequence may be arbitrary. Each such event can be naturally given the probability of 2<sup>−<i>n</i></sup>.

</p><p>These two non-atomic examples are closely related: a sequence <span class="texhtml">(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ...) ∈ {0,1}<sup>∞</sup></span> leads to the number <span class="texhtml">2<sup>−1</sup><i>x</i><sub>1</sub> + 2<sup>−2</sup><i>x</i><sub>2</sub> + ⋯ ∈ [0,1]</span>. This is not a <a href="One-to-one_correspondence" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> between {0,1}<sup>∞</sup> and [0,1] however: it is an <a href="Standard_probability_space" title="Standard probability space">isomorphism modulo zero</a>, which allows for treating the two probability spaces as two forms of the same probability space. In fact, all non-pathological non-atomic probability spaces are the same in this sense. They are so-called <a href="Standard_probability_space" title="Standard probability space">standard probability spaces</a>. Basic applications of probability spaces are insensitive to standardness. However, non-discrete conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure.

</p>

<div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Probability_distribution">Probability distribution</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probability_distribution" title="Probability distribution">Probability distribution</a></div>

<div class="mw-heading mw-heading3"><h3 id="Random_variables">Random variables</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_variable" title="Random variable">Random variable</a></div>

<p>A random variable <i>X</i> is a <a href="Measurable_function" title="Measurable function">measurable function</a> <i>X</i>: Ω → <i>S</i> from the sample space Ω to another measurable space <i>S</i> called the <i>state space</i>.

</p><p>If <i>A</i> ⊂ <i>S</i>, the notation Pr(<i>X</i> ∈ <i>A</i>) is a commonly used shorthand for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}</annotation>

</semantics>

</math></span><img src="./9df6935102515fb62f59cfbe654ad127a639cfda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.6ex; height:2.843ex;" alt="{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Defining_the_events_in_terms_of_the_sample_space">Defining the events in terms of the sample space</h3></div>

<p>If Ω is <a href="Countable" class="mw-redirect" title="Countable">countable</a>, we almost always define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> as the <a href="Power_set" title="Power set">power set</a> of Ω, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> which is trivially a σ-algebra and the biggest one we can create using Ω. We can therefore omit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> and just write (Ω,P) to define the probability space.

</p><p>On the other hand, if Ω is <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> and we use <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=2^{\Omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=2^{\Omega }}</annotation>

</semantics>

</math></span><img src="./b7a6ccebaa8231f5855b17a8c897124bb0713bae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.606ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}=2^{\Omega }}" loading="lazy"></span> we get into trouble defining our probability measure <i>P</i> because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is too "large", i.e. there will often be sets to which it will be impossible to assign a unique measure. In this case, we have to use a smaller σ-algebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>, for example the <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a> of Ω, which is the smallest σ-algebra that makes all open sets measurable.

</p>

<div class="mw-heading mw-heading3"><h3 id="Conditional_probability">Conditional probability</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Conditional_probability" title="Conditional probability">Conditional probability</a></div>

<p>Kolmogorov's definition of probability spaces gives rise to the natural concept of conditional probability. Every set <span class="texhtml mvar" style="font-style:italic;">A</span> with non-zero probability (that is, <span class="texhtml"><i>P</i>(<i>A</i>) > 0</span>) defines another probability measure

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(B\mid A)={P(B\cap A) \over P(A)}}</annotation>

</semantics>

</math></span></span>

on the space. This is usually pronounced as the "probability of <i>B</i> given <i>A</i>".

</p><p>For any event <span class="texhtml"><i>A</i></span> such that <span class="texhtml"><i>P</i>(<i>A</i>) > 0</span>, the function <span class="texhtml"><i>Q</i></span> defined by <span class="texhtml"><i>Q</i>(<i>B</i>) = <i>P</i>(<i>B</i> | <i>A</i>)</span> for all events <span class="texhtml mvar" style="font-style:italic;">B</span> is itself a probability measure.

</p>

<div class="mw-heading mw-heading3"><h3 id="Independence">Independence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">Statistical independence</a></div>

<p>Two events, <i>A</i> and <i>B</i> are said to be independent if <span class="texhtml"><i>P</i>(<i>A</i> ∩ <i>B</i>) = <i>P</i>(<i>A</i>) <i>P</i>(<i>B</i>)</span>.

</p><p>Two random variables, <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">Y</span>, are said to be independent if any event defined in terms of <span class="texhtml mvar" style="font-style:italic;">X</span> is independent of any event defined in terms of <span class="texhtml mvar" style="font-style:italic;">Y</span>. Formally, they generate independent σ-algebras, where two σ-algebras <span class="texhtml mvar" style="font-style:italic;">G</span> and <span class="texhtml mvar" style="font-style:italic;">H</span>, which are subsets of <span class="texhtml mvar" style="font-style:italic;">F</span> are said to be independent if any element of <span class="texhtml mvar" style="font-style:italic;">G</span> is independent of any element of <span class="texhtml mvar" style="font-style:italic;">H</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Mutual_exclusivity">Mutual exclusivity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></div>

<p>Two events, <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> are said to be mutually exclusive or <i>disjoint</i> if the occurrence of one implies the non-occurrence of the other, i.e., their intersection is empty. This is a stronger condition than the probability of their intersection being zero.

</p><p>If <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> are disjoint events, then <span class="texhtml"><i>P</i>(<i>A</i> ∪ <i>B</i>) = <i>P</i>(<i>A</i>) + <i>P</i>(<i>B</i>)</span>. This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if <span class="texhtml mvar" style="font-style:italic;">Z</span> is a <a href="Normal_distribution" title="Normal distribution">normally distributed</a> random variable, then <span class="texhtml"><i>P</i>(<i>Z</i> = <i>x</i>)</span> is 0 for any <span class="texhtml mvar" style="font-style:italic;">x</span>, but <span class="texhtml"><i>P</i>(<i>Z</i> ∈ <b>R</b>) = 1</span>.

</p><p>The event <span class="texhtml"><i>A</i> ∩ <i>B</i></span> is referred to as "<i>A</i> and <i>B</i>", and the event <span class="texhtml"><i>A</i> ∪ <i>B</i></span> as "<i>A</i> or <i>B</i>".

</p>

<ul><li><a href="Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Fuzzy_measure_theory" title="Fuzzy measure theory">Fuzzy measure theory</a></li>

<li><a href="Filtered_probability_space" class="mw-redirect" title="Filtered probability space">Filtered probability space</a></li>

<li><a href="Talagrand's_concentration_inequality" title="Talagrand's concentration inequality">Talagrand's concentration inequality</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Loève, Michel. Probability Theory, Vol 1. New York: D. Van Nostrand Company, 1955.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><a href="Pierre_Simon_de_Laplace" class="mw-redirect" title="Pierre Simon de Laplace">Pierre Simon de Laplace</a> (1812) <i>Analytical Theory of Probability</i></li></ul>

<dl><dd><dl><dd>The first major treatise blending calculus with probability theory, originally in French: <i>Théorie Analytique des Probabilités</i>.</dd></dl></dd></dl>

<ul><li><a href="Andrei_Nikolajevich_Kolmogorov" class="mw-redirect" title="Andrei Nikolajevich Kolmogorov">Andrei Nikolajevich Kolmogorov</a> (1950) <i>Foundations of the Theory of Probability</i></li></ul>

<dl><dd><dl><dd>The modern measure-theoretic foundation of probability theory; the original German version (<i>Grundbegriffe der Wahrscheinlichkeitrechnung</i>) appeared in 1933.</dd></dl></dd></dl>

<ul><li><a href="Harold_Jeffreys" title="Harold Jeffreys">Harold Jeffreys</a> (1939) <i>The Theory of Probability</i></li></ul>

<dl><dd><dl><dd>An empiricist, Bayesian approach to the foundations of probability theory.</dd></dl></dd></dl>

<ul><li><a href="Edward_Nelson" title="Edward Nelson">Edward Nelson</a> (1987) <i>Radically Elementary Probability Theory</i></li></ul>

<dl><dd><dl><dd>Foundations of probability theory based on nonstandard analysis. Downloadable. <a rel="nofollow" class="external free" href="http://www.math.princeton.edu/~nelson/books.html">http://www.math.princeton.edu/~nelson/books.html</a></dd></dl></dd></dl>

<ul><li><a href="Patrick_Billingsley" title="Patrick Billingsley">Patrick Billingsley</a>: <i>Probability and Measure</i>, John Wiley and Sons, New York, Toronto, London, 1979.</li>

<li>Henk Tijms (2004) <i>Understanding Probability </i></li></ul>

<dl><dd><dl><dd>A lively introduction to probability theory for the beginner, Cambridge Univ. Press.</dd></dl></dd></dl>

<ul><li>David Williams (1991) <i>Probability with martingales</i></li></ul>

<dl><dd><dl><dd>An undergraduate introduction to measure-theoretic probability, Cambridge Univ. Press.</dd></dl></dd></dl>

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</style><cite id="CITEREFGut2005" class="citation book cs1">Gut, Allan (2005). <i>Probability: A Graduate Course</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-22833-0</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFSazonov2001" class="citation cs2">Sazonov, V.V. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Probability_space">"Probability space"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=9eaOxgT5ys0">Animation</a> demonstrating probability space of dice</li>

<li><a rel="nofollow" class="external text" href="http://www.math.uah.edu/stat/">Virtual Laboratories in Probability and Statistics</a> (principal author Kyle Siegrist), especially, <a rel="nofollow" class="external text" href="http://www.math.uah.edu/stat/prob">Probability Spaces</a></li>

<li><a rel="nofollow" class="external text" href="http://en.citizendium.org/wiki/Probability_space">Citizendium</a></li>

<li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Probability_space">Complete probability space</a></li>

<li><span class="citation mathworld" id="Reference-Mathworld-Probability_space"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ProbabilitySpace.html">"Probability space"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Measure_theory138" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Measure_theory138" style="font-size:114%;margin:0 4em"><a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_continuity" title="Absolute continuity">Absolute continuity</a> <a href="Absolute_continuity_(measure_theory)" class="mw-redirect" title="Absolute continuity (measure theory)">of measures</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Lp_space" title="Lp space"><i>L</i><sup><i>p</i></sup> spaces</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a>/<a href="Measurable_function" title="Measurable function">function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li>

<li><a href="Baire_set" title="Baire set">Baire set</a></li>

<li><a href="Borel_set" title="Borel set">Borel set</a>

<ul><li><a href="Borel_equivalence_relation" title="Borel equivalence relation">equivalence relation</a></li></ul></li>

<li><a href="Standard_Borel_space" title="Standard Borel space">Borel space</a></li>

<li><a href="Carath%C3%A9odory's_criterion" title="Carathéodory's criterion">Carathéodory's criterion</a></li>

<li><a href="Cylindrical_%CF%83-algebra" title="Cylindrical σ-algebra">Cylindrical σ-algebra</a>

<ul><li><a href="Cylinder_set" title="Cylinder set">Cylinder set</a></li></ul></li>

<li><a href="Dynkin_system" title="Dynkin system">𝜆-system</a></li>

<li><a href="Essential_range" title="Essential range">Essential range</a>

<ul><li><a href="Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">infimum/supremum</a></li></ul></li>

<li><a href="Locally_measurable_set" class="mw-redirect" title="Locally measurable set">Locally measurable</a></li>

<li><a href="Pi-system" title="Pi-system"><span class="texhtml mvar" style="font-style:italic;">π</span>-system</a></li>

<li><a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a></li>

<li><a href="Non-measurable_set" title="Non-measurable set">Non-measurable set</a>

<ul><li><a href="Vitali_set" title="Vitali set">Vitali set</a></li></ul></li>

<li><a href="Support_(measure_theory)" title="Support (measure theory)">Support</a></li>

<li><a href="Transverse_measure" title="Transverse measure">Transverse measure</a></li>

<li><a href="Universally_measurable_set" title="Universally measurable set">Universally measurable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Measure_(mathematics)" title="Measure (mathematics)">measures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atomic_measure" class="mw-redirect" title="Atomic measure">Atomic</a></li>

<li><a href="Baire_measure" title="Baire measure">Baire</a></li>

<li><a href="Banach_measure" title="Banach measure">Banach</a></li>

<li><a href="Besov_measure" title="Besov measure">Besov</a></li>

<li><a href="Borel_measure" title="Borel measure">Borel</a></li>

<li><a href="Brown_measure" title="Brown measure">Brown</a></li>

<li><a href="Complex_measure" title="Complex measure">Complex</a></li>

<li><a href="Complete_measure" title="Complete measure">Complete</a></li>

<li><a href="Content_(measure_theory)" title="Content (measure theory)">Content</a></li>

<li>(<a href="Logarithmically_concave_measure" title="Logarithmically concave measure">Logarithmically</a>) <a href="Convex_measure" title="Convex measure">Convex</a></li>

<li><a href="Decomposable_measure" title="Decomposable measure">Decomposable</a></li>

<li><a href="Discrete_measure" title="Discrete measure">Discrete</a></li>

<li><a href="Equivalence_(measure_theory)" title="Equivalence (measure theory)">Equivalent</a></li>

<li><a href="Finite_measure" title="Finite measure">Finite</a></li>

<li><a href="Inner_measure" title="Inner measure">Inner</a></li>

<li>(<a href="Quasi-invariant_measure" title="Quasi-invariant measure">Quasi-</a>) <a href="Invariant_measure" title="Invariant measure">Invariant</a></li>

<li><a href="Locally_finite_measure" title="Locally finite measure">Locally finite</a></li>

<li><a href="Maximising_measure" title="Maximising measure">Maximising</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer</a></li>

<li><a href="Outer_measure" title="Outer measure">Outer</a></li>

<li><a href="Perfect_measure" title="Perfect measure">Perfect</a></li>

<li><a href="Pre-measure" title="Pre-measure">Pre-measure</a></li>

<li>(<a href="Sub-probability_measure" title="Sub-probability measure">Sub-</a>) <a href="Probability_measure" title="Probability measure">Probability</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li>

<li><a href="Radon_measure" title="Radon measure">Radon</a></li>

<li><a href="Random_measure" title="Random measure">Random</a></li>

<li><a href="Regular_measure" title="Regular measure">Regular</a>

<ul><li><a href="Borel_regular_measure" title="Borel regular measure">Borel regular</a></li>

<li><a href="Inner_regular_measure" class="mw-redirect" title="Inner regular measure">Inner regular</a></li>

<li><a href="Outer_regular_measure" class="mw-redirect" title="Outer regular measure">Outer regular</a></li></ul></li>

<li><a href="Saturated_measure" title="Saturated measure">Saturated</a></li>

<li><a href="Set_function" title="Set function">Set function</a></li>

<li><a href="%CE%A3-finite_measure" title="Σ-finite measure">σ-finite</a></li>

<li><a href="S-finite_measure" title="S-finite measure">s-finite</a></li>

<li><a href="Signed_measure" title="Signed measure">Signed</a></li>

<li><a href="Singular_measure" title="Singular measure">Singular</a></li>

<li><a href="Spectral_measure" class="mw-redirect" title="Spectral measure">Spectral</a></li>

<li><a href="Strictly_positive_measure" title="Strictly positive measure">Strictly positive</a></li>

<li><a href="Tightness_of_measures" title="Tightness of measures">Tight</a></li>

<li><a href="Vector_measure" title="Vector measure">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particular measures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Counting_measure" title="Counting measure">Counting</a></li>

<li><a href="Dirac_measure" title="Dirac measure">Dirac</a></li>

<li><a href="Euler_measure" title="Euler measure">Euler</a></li>

<li><a href="Gaussian_measure" title="Gaussian measure">Gaussian</a></li>

<li><a href="Harmonic_measure" title="Harmonic measure">Harmonic</a></li>

<li><a href="Hausdorff_measure" title="Hausdorff measure">Hausdorff</a></li>

<li><a href="Intensity_measure" title="Intensity measure">Intensity</a></li>

<li><a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a>

<ul><li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Infinite-dimensional</a></li></ul></li>

<li><a href="Positive_real_numbers#Logarithmic_measure" title="Positive real numbers">Logarithmic</a></li>

<li><a href="Product_measure" title="Product measure">Product</a>

<ul><li><a href="Projection_(measure_theory)" title="Projection (measure theory)">Projections</a></li></ul></li>

<li><a href="Pushforward_measure" title="Pushforward measure">Pushforward</a></li>

<li><a href="Spherical_measure" title="Spherical measure">Spherical measure</a></li>

<li><a href="Tangent_measure" title="Tangent measure">Tangent</a></li>

<li><a href="Trivial_measure" title="Trivial measure">Trivial</a></li>

<li><a href="Young_measure" title="Young measure">Young</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Measurable_function" title="Measurable function">Measurable function</a>

<ul><li><a href="Bochner_measurable_function" title="Bochner measurable function">Bochner</a></li>

<li><a href="Strongly_measurable_function" title="Strongly measurable function">Strongly</a></li>

<li><a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a></li></ul></li>

<li>Convergence: <a href="Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">almost everywhere</a></li>

<li><a href="Convergence_of_measures" title="Convergence of measures">of measures</a></li>

<li><a href="Convergence_in_measure" title="Convergence in measure">in measure</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">of random variables</a>

<ul><li><a href="Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">in distribution</a></li>

<li><a href="Convergence_in_probability" class="mw-redirect" title="Convergence in probability">in probability</a></li></ul></li>

<li><a href="Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li>

<li>Random: <a href="Random_compact_set" title="Random compact set">compact set</a></li>

<li><a href="Random_element" title="Random element">element</a></li>

<li><a href="Random_measure" title="Random measure">measure</a></li>

<li><a href="Stochastic_process" title="Stochastic process">process</a></li>

<li><a href="Random_variable" title="Random variable">variable</a></li>

<li><a href="Multivariate_random_variable" title="Multivariate random variable">vector</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carath%C3%A9odory's_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li>

<li>Convergence theorems

<ul><li><a href="Dominated_convergence_theorem" title="Dominated convergence theorem">Dominated</a></li>

<li><a href="Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone</a></li>

<li><a href="Vitali_convergence_theorem" title="Vitali convergence theorem">Vitali</a></li></ul></li>

<li>Decomposition theorems

<li><a href="Jordan_decomposition_theorem" class="mw-redirect" title="Jordan decomposition theorem">Jordan</a></li>

<li><a href="Maharam's_theorem" title="Maharam's theorem">Maharam's</a></li></ul></li>

<li><a href="Egorov's_theorem" title="Egorov's theorem">Egorov's</a></li>

<li><a href="Fatou's_lemma" title="Fatou's lemma">Fatou's lemma</a></li>

<li><a href="Fubini's_theorem" title="Fubini's theorem">Fubini's</a>

<ul><li><a href="Fubini%E2%80%93Tonelli_theorem" class="mw-redirect" title="Fubini–Tonelli theorem">Fubini–Tonelli</a></li></ul></li>

<li><a href="H%C3%B6lder's_inequality" title="Hölder's inequality">Hölder's inequality</a></li>

<li><a href="Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a></li>

<li><a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym</a></li>

<li><a href="Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem" title="Riesz–Markov–Kakutani representation theorem">Riesz–Markov–Kakutani representation theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Disintegration_theorem" title="Disintegration theorem">Disintegration theorem</a>

<ul><li><a href="Lifting_theory" title="Lifting theory">Lifting theory</a></li></ul></li>

<li><a href="Lebesgue's_density_theorem" title="Lebesgue's density theorem">Lebesgue's density theorem</a></li>

<li><a href="Lebesgue_differentiation_theorem" title="Lebesgue differentiation theorem">Lebesgue differentiation theorem</a></li>

<li><a href="Sard's_theorem" title="Sard's theorem">Sard's theorem</a></li>

<li><a href="Vitali%E2%80%93Hahn%E2%80%93Saks_theorem" title="Vitali–Hahn–Saks theorem">Vitali–Hahn–Saks theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size: 85%;">For <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li>

<li><a href="Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a>

<ul><li><a href="Milman's_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li>

<li><a href="Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li>

<li><a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li>

<li><a href="Vitale's_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_analysis" title="Convex analysis">Convex analysis</a></li>

<li><a href="Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Spectral_theory" title="Spectral theory">Spectral theory</a></li></ul>

</div></td></tr></tbody></table></div>

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