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  skos:prefLabel "calcul intégral"@fr, "integral calculus"@en ;
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  skos:definition """En mathématiques, et plus précisément en analyse, le <b>théorème de Fubini</b> fournit des informations sur le calcul d'intégrales définies sur des ensembles produits et permet le calcul de telles intégrales. Ce résultat a été introduit par Guido Fubini en 1907. Il indique que sous certaines conditions, pour intégrer une fonction à plusieurs variables, on peut intégrer les variables les unes à la suite des autres. On peut changer l'ordre d'intégration si l'intégrable double de la valeur absolue de la fonction est finie&nbsp;:
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ight){\\	ext{d}}x=\\\\int _{Y}\\\\left(\\\\int _{X}f(x,y)\\\\,{\\	ext{d}}x\\
ight){\\	ext{d}}y\\\\qquad {\\	ext{ si }}\\\\qquad \\\\iint \\\\limits _{X\\	imes Y}|f(x,y)|\\\\,{\\	ext{d}}(x,y)<+\\\\infty .}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\,\\\\iint \\\\limits _{X\\	imes Y}f(x,y)\\\\,{\\	ext{d}}(x,y)=\\\\int _{X}\\\\left(\\\\int _{Y}f(x,y)\\\\,{\\	ext{d}}y\\
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ight){\\	ext{d}}y\\\\qquad {\\	ext{ si }}\\\\qquad \\\\iint \\\\limits _{X\\	imes Y}|f(x,y)|\\\\,{\\	ext{d}}(x,y)&lt;+\\\\infty .}</annotation>
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ight){\\	ext{d}}x=\\\\int _{Y}\\\\left(\\\\int _{X}f(x,y)\\\\,{\\	ext{d}}x\\
ight){\\	ext{d}}y\\\\qquad {\\	ext{ si }}\\\\qquad \\\\iint \\\\limits _{X\\	imes Y}|f(x,y)|\\\\,{\\	ext{d}}(x,y)<+\\\\infty .}"></div>
<br/>

<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fubini">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Fubini</a>)"""@fr, """In mathematical analysis, <b>Fubini's theorem</b> is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.
<br/><div class="mwe-math-element"><div 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 \\\\,\\\\iint \\\\limits _{X\\	imes Y}f(x,y)\\\\,{\\	ext{d}}(x,y)=\\\\int _{X}\\\\left(\\\\int _{Y}f(x,y)\\\\,{\\	ext{d}}y\\
ight){\\	ext{d}}x=\\\\int _{Y}\\\\left(\\\\int _{X}f(x,y)\\\\,{\\	ext{d}}x\\
ight){\\	ext{d}}y\\\\qquad {\\	ext{ if }}\\\\qquad \\\\iint \\\\limits _{X\\	imes Y}|f(x,y)|\\\\,{\\	ext{d}}(x,y)<+\\\\infty .}">
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<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d428f0c71bcde33a85337f71dbb3c1cfdff947" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.155ex; width:108.165ex; height:7.676ex;" alt="{\\\\displaystyle \\\\,\\\\iint \\\\limits _{X\\	imes Y}f(x,y)\\\\,{\\	ext{d}}(x,y)=\\\\int _{X}\\\\left(\\\\int _{Y}f(x,y)\\\\,{\\	ext{d}}y\\
ight){\\	ext{d}}x=\\\\int _{Y}\\\\left(\\\\int _{X}f(x,y)\\\\,{\\	ext{d}}x\\
ight){\\	ext{d}}y\\\\qquad {\\	ext{ if }}\\\\qquad \\\\iint \\\\limits _{X\\	imes Y}|f(x,y)|\\\\,{\\	ext{d}}(x,y)<+\\\\infty .}"></div>
<br/>Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. <b>Tonelli's theorem</b>, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains.
<br/>A related theorem is often called <b>Fubini's theorem for infinite series</b>, which states that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\	extstyle \\\\{a_{m,n}\\\\}_{m=1,n=1}^{\\\\infty }}">
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{(m,n)\\\\in \\\\mathbb {N} \\	imes \\\\mathbb {N} }a_{m,n}=\\\\sum _{m=1}^{\\\\infty }\\\\sum _{n=1}^{\\\\infty }a_{m,n}=\\\\sum _{n=1}^{\\\\infty }\\\\sum _{m=1}^{\\\\infty }a_{m,n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c91fbe96f4af8cfd53c23ec92b9c62361b00134" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:43.888ex; height:7.343ex;" alt="{\\\\displaystyle \\\\sum _{(m,n)\\\\in \\\\mathbb {N} \\	imes \\\\mathbb {N} }a_{m,n}=\\\\sum _{m=1}^{\\\\infty }\\\\sum _{n=1}^{\\\\infty }a_{m,n}=\\\\sum _{n=1}^{\\\\infty }\\\\sum _{m=1}^{\\\\infty }a_{m,n}}"></span>
<br/>Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series.  In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Fubini%27s_theorem">https://en.wikipedia.org/wiki/Fubini%27s_theorem</a>)"""@en ;
  a skos:Concept ;
  skos:broader psr:-BLP2HLSP-6 ;
  dc:created "2023-08-02"^^xsd:date ;
  skos:prefLabel "théorème de Fubini"@fr, "Fubini's theorem"@en ;
  skos:inScheme psr: ;
  dc:modified "2023-08-02"^^xsd:date .