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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Absolutely integrable function</span></span>
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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"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>absolutely integrable function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="Absolute_value" title="Absolute value">absolute value</a> is <a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">integrable</a>, meaning that the integral of the absolute value over the whole <a href="Domain_of_a_function" title="Domain of a function">domain</a> is finite.
</p><p>For a <a href="Real_number" title="Real number">real</a>-valued function, since
<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 \int |f(x)|\,dx=\int f^{+}(x)\,dx+\int f^{-}(x)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>+</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int |f(x)|\,dx=\int f^{+}(x)\,dx+\int f^{-}(x)\,dx}</annotation>
</semantics>
</math></span></span>
where
<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 f^{+}(x)=\max(f(x),0),\ \ \ f^{-}(x)=\max(-f(x),0),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mtext> </mtext>
<mtext> </mtext>
<mtext> </mtext>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{+}(x)=\max(f(x),0),\ \ \ f^{-}(x)=\max(-f(x),0),}</annotation>
</semantics>
</math></span></span>
</p><p>both <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 \int f^{+}(x)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \int f^{+}(x)\,dx}</annotation>
</semantics>
</math></span><img src="./eb8b8135aa5dc915bcf833fb5ecf9c2081ff7f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.708ex; height:3.176ex;" alt="{\textstyle \int f^{+}(x)\,dx}" 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="{\textstyle \int f^{-}(x)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \int f^{-}(x)\,dx}</annotation>
</semantics>
</math></span><img src="./4ba7251f1e50df13563a989d93ab631156471a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.708ex; height:3.176ex;" alt="{\textstyle \int f^{-}(x)\,dx}" loading="lazy"></span> must be finite. In <a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a>, this is exactly the requirement for any <a href="Measurable_function" title="Measurable function">measurable function</a> <i>f</i> to be considered integrable, with the integral then equaling <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 \int f^{+}(x)\,dx-\int f^{-}(x)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \int f^{+}(x)\,dx-\int f^{-}(x)\,dx}</annotation>
</semantics>
</math></span><img src="./a8a519ded9b58490a8925ca635657b6f5b8778ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.256ex; height:3.176ex;" alt="{\textstyle \int f^{+}(x)\,dx-\int f^{-}(x)\,dx}" loading="lazy"></span>, so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions.
</p><p>The same thing goes for a <a href="Complex_number" title="Complex number">complex</a>-valued function. Let us define
<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 f^{+}(x)=\max(\Re f(x),0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{+}(x)=\max(\Re f(x),0)}</annotation>
</semantics>
</math></span></span>
<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 f^{-}(x)=\max(-\Re f(x),0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{-}(x)=\max(-\Re f(x),0)}</annotation>
</semantics>
</math></span></span>
<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 f^{+i}(x)=\max(\Im f(x),0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
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<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{+i}(x)=\max(\Im f(x),0)}</annotation>
</semantics>
</math></span></span>
<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 f^{-i}(x)=\max(-\Im f(x),0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">ℑ<!-- ℑ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{-i}(x)=\max(-\Im f(x),0)}</annotation>
</semantics>
</math></span></span>
where <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 \Re f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℜ<!-- ℜ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Re f(x)}</annotation>
</semantics>
</math></span><img src="./7203b96a226e0ace198a436a956d9356890ccf6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.342ex; height:2.843ex;" alt="{\displaystyle \Re f(x)}" 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 \Im f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℑ<!-- ℑ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Im f(x)}</annotation>
</semantics>
</math></span><img src="./336b2be7f5b7c426405dcb2fec55e9e92b12b762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.705ex; height:2.843ex;" alt="{\displaystyle \Im f(x)}" loading="lazy"></span> are the <a href="Complex_number" title="Complex number">real and imaginary parts</a> 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 f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span>. Then
<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 |f(x)|\leq f^{+}(x)+f^{-}(x)+f^{+i}(x)+f^{-i}(x)\leq {\sqrt {2}}\,|f(x)|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>i</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq f^{+}(x)+f^{-}(x)+f^{+i}(x)+f^{-i}(x)\leq {\sqrt {2}}\,|f(x)|}</annotation>
</semantics>
</math></span></span>
so
<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 \int |f(x)|\,dx\leq \int f^{+}(x)\,dx+\int f^{-}(x)\,dx+\int f^{+i}(x)\,dx+\int f^{-i}(x)\,dx\leq {\sqrt {2}}\int |f(x)|\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>≤<!-- ≤ --></mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>+</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>+</mo>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
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<annotation encoding="application/x-tex">{\displaystyle \int |f(x)|\,dx\leq \int f^{+}(x)\,dx+\int f^{-}(x)\,dx+\int f^{+i}(x)\,dx+\int f^{-i}(x)\,dx\leq {\sqrt {2}}\int |f(x)|\,dx}</annotation>
</semantics>
</math></span></span>
This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for the function to be "Lebesgue integrable".
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Absolutely_integrable_function">"Absolutely integrable function – Encyclopedia of Mathematics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">9 October</span> 2015</span>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><a href="Terence_Tao" title="Terence Tao">Tao, Terence</a>, <i>Analysis 2</i>, 3rd ed., Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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