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Transition kernel

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In the <a href="Probability_theory" title="Probability theory">mathematics of probability</a>, a transition kernel or kernel is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> in mathematics that has different applications. Kernels can for example be used to define <a href="Random_measure" title="Random measure">random measures</a> or <a href="Stochastic_process" title="Stochastic process">stochastic processes</a>. The most important example of kernels are the <a href="Markov_kernel" title="Markov kernel">Markov kernels</a>.

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

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,{\mathcal {S}})}">

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

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</math><img src="./0d6e0bcf0a35935092ac72ee8eaeeb3eea8063c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.835ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {S}})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,{\mathcal {T}})}">

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

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</math><img src="./f5cfabf9211b285a28f9b9c2b517d8ff0985206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (T,{\mathcal {T}})}" loading="lazy"> be two <a href="Measurable_space" title="Measurable space">measurable spaces</a>. A function

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<annotation encoding="application/x-tex">{\displaystyle \kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]}</annotation>

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</math><img src="./f2f6d48846ae254d72ea28a3cab412fc1cdc64ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.884ex; height:2.843ex;" alt="{\displaystyle \kappa \colon S\times {\mathcal {T}}\to [0,+\infty ]}" loading="lazy"></dd></dl>

is called a (transition) kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> if the following two conditions hold:<a href="#cite_note-Klenke180-1">[1]</a>

<ul><li>For any fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {T}}}">

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

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</math><img src="./c89daf67901e329a2ca826b98ebaaee5c8cb8661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.343ex;" alt="{\displaystyle B\in {\mathcal {T}}}" loading="lazy">, the mapping</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle s\mapsto \kappa (s,B)}</annotation>

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</math><img src="./e5a62c172fcc56f9537ca99fdf2c1b9331158a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.741ex; height:2.843ex;" alt="{\displaystyle s\mapsto \kappa (s,B)}" loading="lazy"></dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}/{\mathcal {B}}([0,+\infty ])}</annotation>

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</math><img src="./7444a2cbc5b5dbf170b17c7f68a135c00252c75e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.629ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}/{\mathcal {B}}([0,+\infty ])}" loading="lazy">-<a href="Measurable" class="mw-redirect" title="Measurable">measurable</a>;</dd></dl>

<ul><li>For every fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

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

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</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy">, the mapping</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle B\mapsto \kappa (s,B)}</annotation>

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</math><img src="./400a8fa0e64a6f8aec4e19ae592aac34bd37037a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.415ex; height:2.843ex;" alt="{\displaystyle B\mapsto \kappa (s,B)}" loading="lazy"></dd></dl></dd>

<dd>is a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,{\mathcal {T}})}">

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

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</math><img src="./f5cfabf9211b285a28f9b9c2b517d8ff0985206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (T,{\mathcal {T}})}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Classification_of_transition_kernels">Classification of transition kernels</h2></div>

Transition kernels are usually classified by the measures they define. Those measures are defined as

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<annotation encoding="application/x-tex">{\displaystyle \kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]}</annotation>

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</math><img src="./f2c9e70c6ccefbfe3b000ff697c3b6ed37bd8705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.548ex; height:2.843ex;" alt="{\displaystyle \kappa _{s}\colon {\mathcal {T}}\to [0,+\infty ]}" loading="lazy"></dd></dl>

with

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<annotation encoding="application/x-tex">{\displaystyle \kappa _{s}(B)=\kappa (s,B)}</annotation>

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</math><img src="./bd6387643ac756d4d4bf68c1ad6634b631519ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.051ex; height:2.843ex;" alt="{\displaystyle \kappa _{s}(B)=\kappa (s,B)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {T}}}">

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

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</math><img src="./c89daf67901e329a2ca826b98ebaaee5c8cb8661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.343ex;" alt="{\displaystyle B\in {\mathcal {T}}}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

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

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</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy">. With this notation, the kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> is called<a href="#cite_note-Klenke180-1">[1]</a><a href="#cite_note-Kallenberg30-2">[2]</a>

<ul><li>a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}}">

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

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</math><img src="./4fe85dce588b6087680d91fb9ecdbceb1930e82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.009ex;" alt="{\displaystyle \kappa _{s}}" loading="lazy"> are <a href="Sub-probability_measure" title="Sub-probability measure">sub-probability measures</a></li>

<li>a <a href="Markov_kernel" title="Markov kernel">Markov kernel</a>, stochastic kernel or probability kernel if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}}">

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

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</math><img src="./4fe85dce588b6087680d91fb9ecdbceb1930e82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.009ex;" alt="{\displaystyle \kappa _{s}}" loading="lazy"> are <a href="Probability_measure" title="Probability measure">probability measures</a></li>

<li>a finite kernel if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}}">

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

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</math><img src="./4fe85dce588b6087680d91fb9ecdbceb1930e82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.009ex;" alt="{\displaystyle \kappa _{s}}" loading="lazy"> are <a href="Finite_measure" title="Finite measure">finite measures</a></li>

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-finite kernel if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}}">

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

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</math><img src="./4fe85dce588b6087680d91fb9ecdbceb1930e82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.009ex;" alt="{\displaystyle \kappa _{s}}" loading="lazy"> are <a href="Sigma-finite_measure" class="mw-redirect" title="Sigma-finite measure"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-finite measures</a></li>

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">-finite kernel if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> can be written as a countable sum of finite kernels (so that in particular, all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}}">

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

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</math><img src="./4fe85dce588b6087680d91fb9ecdbceb1930e82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.009ex;" alt="{\displaystyle \kappa _{s}}" loading="lazy"> are <a href="S-finite_measure" title="S-finite measure"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">-finite measures</a>).</li>

<li>a uniformly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-finite kernel if there are at most countably many measurable sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1},B_{2},\dots }">

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

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</math><img src="./1c7080b632c5947668866dc110cb674512b8223c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.428ex; height:2.509ex;" alt="{\displaystyle B_{1},B_{2},\dots }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{s}(B_{i})<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa _{s}(B_{i})<\infty }</annotation>

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</math><img src="./321f6fa0cf920310fe7e79f707c06095048cea21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.137ex; height:2.843ex;" alt="{\displaystyle \kappa _{s}(B_{i})<\infty }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

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

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</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle i\in \mathbb {N} }</annotation>

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</math><img src="./e64c8c5906eb3eb9d7a8b1ed1e31de4e5fc6c632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {N} }" loading="lazy">.</li></ul>

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

In this section, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,{\mathcal {S}})}">

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

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</math><img src="./0d6e0bcf0a35935092ac72ee8eaeeb3eea8063c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.835ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {S}})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,{\mathcal {T}})}">

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

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</math><img src="./f5cfabf9211b285a28f9b9c2b517d8ff0985206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (T,{\mathcal {T}})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,{\mathcal {U}})}">

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

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</math><img src="./4bec7dc698184b2c822405b95b4fa7fc92f065b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.223ex; height:2.843ex;" alt="{\displaystyle (U,{\mathcal {U}})}" loading="lazy"> be measurable spaces and denote the <a href="Product_%CF%83-algebra" class="mw-redirect" title="Product σ-algebra">product σ-algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

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

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</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">

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

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</math><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}\otimes {\mathcal {T}}}">

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

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</math><img src="./8d31ec038f3dd5b204808c130d60cadbde92ad60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.269ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}\otimes {\mathcal {T}}}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Product_of_kernels">Product of kernels</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definition_2">Definition</h4></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}}">

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

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</math><img src="./a774b0655476cb71ea741372953bac106cafcb53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{1}}" loading="lazy"> be a s-finite kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{2}}">

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

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</math><img src="./fc389bc005086e290488b517e85d2d0a52db5b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{2}}" loading="lazy"> be a s-finite kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\times T}">

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

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</math><img src="./7d25dd1726c95ab55a9812e09f4a7610cbf25b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.976ex; height:2.176ex;" alt="{\displaystyle S\times T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Then the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}\otimes \kappa ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa ^{1}\otimes \kappa ^{2}}</annotation>

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</math><img src="./24712096c12220cf0bccb83daae0bb8df7053a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle \kappa ^{1}\otimes \kappa ^{2}}" loading="lazy"> of the two kernels is defined as<a href="#cite_note-Kallenberg33-3">[3]</a><a href="#cite_note-Klenke279-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]}</annotation>

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</math><img src="./51d3de728ea12c25432c0e4ecd06b73a4159d6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.611ex; height:3.176ex;" alt="{\displaystyle \kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)}</annotation>

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</math><img src="./1701d5e5f7b0a389fd6c1986d5564a78f4f3da7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.935ex; height:5.676ex;" alt="{\displaystyle \kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {T}}\otimes {\mathcal {U}}}">

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

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</math><img src="./833f2782ed5a6e2554d75c70de33b5b159aa7579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.957ex; height:2.509ex;" alt="{\displaystyle A\in {\mathcal {T}}\otimes {\mathcal {U}}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Properties_and_comments">Properties and comments</h4></div>

The product of two kernels is a kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\times U}">

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

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</math><img src="./d926c4e85e064bb42650cad0b5cea2b32ded12a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.259ex; height:2.176ex;" alt="{\displaystyle T\times U}" loading="lazy">. It is again a s-finite kernel and is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-finite kernel if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}}">

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

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</math><img src="./a774b0655476cb71ea741372953bac106cafcb53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{2}}">

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

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</math><img src="./fc389bc005086e290488b517e85d2d0a52db5b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{2}}" loading="lazy"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">-finite kernels. The product of kernels is also <a href="Associative" class="mw-redirect" title="Associative">associative</a>, meaning it satisfies

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<annotation encoding="application/x-tex">{\displaystyle (\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})}</annotation>

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</math><img src="./e1af65b83e0658f58341ccecd5b8e1a2ee0cca3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.438ex; height:3.176ex;" alt="{\displaystyle (\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})}" loading="lazy"></dd></dl>

for any three suitable s-finite kernels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}">

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

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</math><img src="./66ee5eefb2e3791fabfc075a4982a71bde787f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.248ex; height:3.009ex;" alt="{\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}" loading="lazy">.

The product is also well-defined if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{2}}">

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

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</math><img src="./fc389bc005086e290488b517e85d2d0a52db5b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{2}}" loading="lazy"> is a kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. In this case, it is treated like a kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\times T}">

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

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</math><img src="./7d25dd1726c95ab55a9812e09f4a7610cbf25b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.976ex; height:2.176ex;" alt="{\displaystyle S\times T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> that is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. This is equivalent to setting

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<annotation encoding="application/x-tex">{\displaystyle \kappa ((s,t),A):=\kappa (t,A)}</annotation>

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</math><img src="./63a7a5dbd7ab58ca8af404402dbd32db58dbfa13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.209ex; height:2.843ex;" alt="{\displaystyle \kappa ((s,t),A):=\kappa (t,A)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {U}}}">

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

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</math><img src="./a6b6f37ca99e14f036b404c5a6f2c48f75b7bd10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.181ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {U}}}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

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

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</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy">.<a href="#cite_note-Klenke279-4">[4]</a><a href="#cite_note-Kallenberg33-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Composition_of_kernels">Composition of kernels</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definition_3">Definition</h4></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}}">

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</math><img src="./a774b0655476cb71ea741372953bac106cafcb53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{1}}" loading="lazy"> be a s-finite kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{2}}">

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

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</math><img src="./fc389bc005086e290488b517e85d2d0a52db5b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{2}}" loading="lazy"> a s-finite kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\times T}">

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

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</math><img src="./7d25dd1726c95ab55a9812e09f4a7610cbf25b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.976ex; height:2.176ex;" alt="{\displaystyle S\times T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Then the composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}\cdot \kappa ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa ^{1}\cdot \kappa ^{2}}</annotation>

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</math><img src="./ba678d712cba635c8fbfad17eab366233980187b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.466ex; height:2.676ex;" alt="{\displaystyle \kappa ^{1}\cdot \kappa ^{2}}" loading="lazy"> of the two kernels is defined as<a href="#cite_note-Klenke281-5">[5]</a><a href="#cite_note-Kallenberg33-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle \kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]}</annotation>

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</math><img src="./e3103d9518c0ece3de6fbf016f6972d5d2b984dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.864ex; height:3.176ex;" alt="{\displaystyle \kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle (s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)}</annotation>

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</math><img src="./2136e99cc89077223bfe4a3c4872d0d7155495a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.986ex; height:5.676ex;" alt="{\displaystyle (s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

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

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</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {U}}}">

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<div class="mw-heading mw-heading4"><h4 id="Properties_and_comments_2">Properties and comments</h4></div>

The composition is a kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> that is again s-finite. The composition of kernels is <a href="Associative" class="mw-redirect" title="Associative">associative</a>, meaning it satisfies

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<annotation encoding="application/x-tex">{\displaystyle (\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})}</annotation>

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</math><img src="./919482e92ca90d3837bcca0ceb6f750683da1e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.793ex; height:3.176ex;" alt="{\displaystyle (\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})}" loading="lazy"></dd></dl>

for any three suitable s-finite kernels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}">

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

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</math><img src="./66ee5eefb2e3791fabfc075a4982a71bde787f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.248ex; height:3.009ex;" alt="{\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}" loading="lazy">. Just like the product of kernels, the composition is also well-defined if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{2}}">

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

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</math><img src="./fc389bc005086e290488b517e85d2d0a52db5b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle \kappa ^{2}}" loading="lazy"> is a kernel from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

An alternative notation is for the composition is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ^{1}\kappa ^{2}}">

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

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</math><img src="./2da6aad3d75c6a7d1d4c701b9d365230f5f8482f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.786ex; height:2.676ex;" alt="{\displaystyle \kappa ^{1}\kappa ^{2}}" loading="lazy"><a href="#cite_note-Kallenberg33-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Kernels_as_operators">Kernels as operators</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}}</annotation>

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</math><img src="./508291452ffbc5a95b3158af57c75fd2f2864457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.71ex; height:2.843ex;" alt="{\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}}" loading="lazy"> be the set of positive measurable functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})}">

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<annotation encoding="application/x-tex">{\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})}</annotation>

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</math><img src="./8e3be3dab1b3a28cb3f3a15b3ba12b8c6e83e516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.284ex; height:2.843ex;" alt="{\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})}" loading="lazy">.

Every kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> can be associated with a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a>

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<annotation encoding="application/x-tex">{\displaystyle A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}}</annotation>

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</math><img src="./137b4807a007ec4b7500ac2c5cc3857ec4908f91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.246ex; height:2.843ex;" alt="{\displaystyle A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}}" loading="lazy"></dd></dl>

given by<a href="#cite_note-Kallenberg2930-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle (A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).}</annotation>

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</math><img src="./d3ec9e5580d64f1913c10ffec151efd363b7f8cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.701ex; height:5.676ex;" alt="{\displaystyle (A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).}" loading="lazy"></dd></dl>

The composition of these operators is compatible with the composition of kernels, meaning<a href="#cite_note-Kallenberg33-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}}</annotation>

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</math><img src="./91422f1092ae6ed04a7ee4733252ae08ccaea418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.595ex; height:2.676ex;" alt="{\displaystyle A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}}" loading="lazy"></dd></dl>

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

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

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

</style><cite id="CITEREFKlenke2008" class="citation book cs1">Klenke, Achim (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341">Probability Theory</a>. Berlin: Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341/page/n186">180</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-048-3">10.1007/978-1-84800-048-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-047-6</bdi>.</cite>

</li>

<li id="cite_note-Kallenberg30-2"><a href="#cite_ref-Kallenberg30_2-0">^</a> <cite id="CITEREFKallenberg2017" class="citation book cs1"><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 30. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-41598-7">10.1007/978-3-319-41598-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-41596-3</bdi>.</cite>

</li>

<li id="cite_note-Kallenberg33-3">^ <a href="#cite_ref-Kallenberg33_3-0">a</a> <a href="#cite_ref-Kallenberg33_3-1">b</a> <a href="#cite_ref-Kallenberg33_3-2">c</a> <a href="#cite_ref-Kallenberg33_3-3">d</a> <a href="#cite_ref-Kallenberg33_3-4">e</a> <cite id="CITEREFKallenberg2017" class="citation book cs1"><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 33. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-41598-7">10.1007/978-3-319-41598-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-41596-3</bdi>.</cite>

</li>

<li id="cite_note-Klenke279-4">^ <a href="#cite_ref-Klenke279_4-0">a</a> <a href="#cite_ref-Klenke279_4-1">b</a> <cite id="CITEREFKlenke2008" class="citation book cs1">Klenke, Achim (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341">Probability Theory</a>. Berlin: Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341/page/n280">279</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-048-3">10.1007/978-1-84800-048-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-047-6</bdi>.</cite>

</li>

<li id="cite_note-Klenke281-5"><a href="#cite_ref-Klenke281_5-0">^</a> <cite id="CITEREFKlenke2008" class="citation book cs1">Klenke, Achim (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341">Probability Theory</a>. Berlin: Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitytheor00klen_341/page/n282">281</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-048-3">10.1007/978-1-84800-048-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-047-6</bdi>.</cite>

</li>

<li id="cite_note-Kallenberg2930-6"><a href="#cite_ref-Kallenberg2930_6-0">^</a> <cite id="CITEREFKallenberg2017" class="citation book cs1"><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (2017). Random Measures, Theory and Applications. Switzerland: Springer. pp. 29–30. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-41598-7">10.1007/978-3-319-41598-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-41596-3</bdi>.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

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