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<title>Bochner measurable function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bochner measurable 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> – specifically, in <a href="Functional_analysis" title="Functional analysis">functional analysis</a> – a <b>Bochner-measurable function</b> taking values in a <a href="Banach_space" title="Banach space">Banach space</a> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,

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<mtext> for almost every </mtext>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,}</annotation>

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</math></span><img src="./155ee1449a59a545d5bb4d72c691c5ab5f53a9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.834ex; height:3.843ex;" alt="{\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,}" loading="lazy"></span></dd></dl>

<p>where the functions <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_{n}}">

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

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</math></span><img src="./b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"></span> each have a countable range and for which the pre-image <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_{n}^{-1}(\{x\})}">

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<annotation encoding="application/x-tex">{\displaystyle f_{n}^{-1}(\{x\})}</annotation>

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</math></span><img src="./c6d6dacfa2530180819677ba3e301bc48906332e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.117ex; height:3.176ex;" alt="{\displaystyle f_{n}^{-1}(\{x\})}" loading="lazy"></span> is measurable for each element <i>x</i>. The concept is named after <a href="Salomon_Bochner" title="Salomon Bochner">Salomon Bochner</a>.

</p><p>Bochner-measurable functions are sometimes called <b><a href="Strongly_measurable" class="mw-redirect" title="Strongly measurable">strongly measurable</a></b>, <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 \mu }">

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

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</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>-measurable</b> or just <b>measurable</b> (or <b>uniformly measurable</b> in case that the Banach space is the space of continuous <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> between Banach spaces).

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

<p>The relationship between measurability and weak measurability is given by the following result, known as <b><a href="B._J._Pettis" class="mw-redirect" title="B. J. Pettis">Pettis</a>' theorem</b> or <b>Pettis measurability theorem</b>.

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<p>Function <i>f</i> is <b><a href="Almost_surely" title="Almost surely">almost surely</a> separably valued</b> (or <b>essentially separably valued</b>) if there exists a subset <i>N</i> ⊆ <i>X</i> with <i>μ</i>(<i>N</i>) = 0 such that <i>f</i>(<i>X</i> \ <i>N</i>) ⊆ <i>B</i> is separable.

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<p>A function f  : <i>X</i> → <i>B</i> defined on a <a href="Measure_space" title="Measure space">measure space</a> (<i>X</i>, Σ, <i>μ</i>) and taking values in a Banach space <i>B</i> is (strongly) measurable (with respect to Σ and the <a href="Borel_algebra" class="mw-redirect" title="Borel algebra">Borel algebra</a> on <i>B</i>) <a href="If_and_only_if" title="If and only if">if and only if</a> it is both <a href="Weakly_measurable_function" title="Weakly measurable function">weakly measurable</a> and almost surely separably valued.

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<p>In the case that <i>B</i> is separable, since any subset of a separable Banach space is itself separable, one can take <i>N</i> above to be empty, and it follows that the notions of weak and strong measurability agree when <i>B</i> is separable.

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<ul><li><a href="Bochner_integral" title="Bochner integral">Bochner integral</a> – Concept in mathematics</li>

<li><a href="Bochner_space" title="Bochner space">Bochner space</a> – Type of topological space</li>

<li><a href="Measurable_function" title="Measurable function">Measurable function</a> – Kind of mathematical function</li>

<li><a href="Measurable_space" title="Measurable space">Measurable space</a> – Basic object in measure theory; set and a sigma-algebra</li>

<li><a href="Pettis_integral" title="Pettis integral">Pettis integral</a></li>

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

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

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

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

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</style><cite id="CITEREFShowalter1997" class="citation book cs1">Showalter, Ralph E. (1997). "Theorem III.1.1". <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/monotoneoperatio00show"><i>Monotone operators in Banach space and nonlinear partial differential equations</i></a></span>. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. <a rel="nofollow" class="external text" href="https://archive.org/details/monotoneoperatio00show/page/n109">103</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-0500-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1422252">1422252</a>.</cite>.</li></ul>

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</style><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</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="Banach_space" title="Banach space">Banach</a></li>

<li><a href="Besov_space" title="Besov space">Besov</a></li>

<li><a href="Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li>

<li><a href="Hilbert_space" title="Hilbert space">Hilbert</a></li>

<li><a href="H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li>

<li><a href="Nuclear_space" title="Nuclear space">Nuclear</a></li>

<li><a href="Orlicz_space" title="Orlicz space">Orlicz</a></li>

<li><a href="Schwartz_space" title="Schwartz space">Schwartz</a></li>

<li><a href="Sobolev_space" title="Sobolev space">Sobolev</a></li>

<li><a href="Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</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="Barrelled_space" title="Barrelled space">Barrelled</a></li>

<li><a href="Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li>

<li><a href="Dual_space" title="Dual space">Dual</a> (<a href="Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li>

<li><a href="Reflexive_space" title="Reflexive space">Reflexive</a></li>

<li><a href="Separable_space" title="Separable space">Separable</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</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="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li>

<li><a href="Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li>

<li><a href="Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li>

<li><a href="Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li>

<li><a href="Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li>

<li><a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li>

<li><a href="Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</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="Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded</a></li>

<li><a href="Compact_operator" title="Compact operator">Compact</a></li>

<li><a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li>

<li><a href="Normal_operator" title="Normal operator">Normal</a></li>

<li><a href="Nuclear_operator" title="Nuclear operator">Nuclear</a></li>

<li><a href="Trace_class" title="Trace class">Trace class</a></li>

<li><a href="Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li>

<li><a href="Unbounded_operator" title="Unbounded operator">Unbounded</a></li>

<li><a href="Unitary_operator" title="Unitary operator">Unitary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</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="Banach_algebra" title="Banach algebra">Banach algebra</a></li>

<li><a href="C*-algebra" title="C*-algebra">C*-algebra</a></li>

<li><a href="Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li>

<li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li>

<li><a href="Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</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="Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li>

<li><a href="Mahler's_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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="Hardy_space" title="Hardy space">Hardy space</a></li>

<li><a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li>

<li><a href="Heat_kernel" title="Heat kernel">Heat kernel</a></li>

<li><a href="Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li>

<li><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></li>

<li><a href="Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</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="Approximation_property" title="Approximation property">Approximation property</a></li>

<li><a href="Balanced_set" title="Balanced set">Balanced set</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Weak_topology" title="Weak topology">Weak topology</a></li>

<li><a href="Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li>

<li><a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Analysis_in_topological_vector_spaces159" 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="Analysis_in_topological_vector_spaces159" style="font-size:114%;margin:0 4em"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a> in <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</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="Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul></li>

<li><a href="Bochner_space" title="Bochner space">Bochner space</a></li>

<li><a href="Convex_series" title="Convex series">Convex series</a></li>

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

<li><a href="Infinite-dimensional_vector_function" title="Infinite-dimensional vector function">Infinite-dimensional vector function</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Derivative" title="Derivative">Derivatives</a></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="Differentiable_vector-valued_functions_from_Euclidean_space" title="Differentiable vector-valued functions from Euclidean space">Differentiable vector-valued functions from Euclidean space</a></li>

<li><a href="Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li>

<li><a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a>

<ul><li><a href="Total_derivative" title="Total derivative">Total</a></li></ul></li>

<li><a href="Functional_derivative" title="Functional derivative">Functional derivative</a></li>

<li><a href="Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a>

<ul><li><a href="Directional_derivative" title="Directional derivative">Directional</a></li></ul></li>

<li><a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">Generalizations of the derivative</a></li>

<li><a href="Hadamard_derivative" title="Hadamard derivative">Hadamard derivative</a></li>

<li><a href="Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">Holomorphic</a></li>

<li><a href="Quasi-derivative" title="Quasi-derivative">Quasi-derivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurability</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="Besov_measure" title="Besov measure">Besov measure</a></li>

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

<ul><li><a href="Canonical_Gaussian_cylinder_set_measure" class="mw-redirect" title="Canonical Gaussian cylinder set measure">Canonical Gaussian</a></li>

<li><a href="Classical_Wiener_measure" class="mw-redirect" title="Classical Wiener measure">Classical Wiener measure</a></li></ul></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a> like <a href="Set_function" title="Set function">set functions</a>

<ul><li><a href="Gaussian_measure#Infinite-dimensional_spaces" title="Gaussian measure">infinite-dimensional Gaussian measure</a></li>

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

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

<li> / <a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="Strongly_measurable_function" title="Strongly measurable function">Strongly</a> <a href="Measurable_function" title="Measurable function">measurable function</a></li>

<li><a href="Radonifying_function" title="Radonifying function">Radonifying function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral" title="Integral">Integrals</a></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="Bochner_integral" title="Bochner integral">Bochner</a></li>

<li><a href="Direct_integral" title="Direct integral">Direct integral</a></li>

<li><a href="Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li>

<li><a href="Pettis_integral" title="Pettis integral">Gelfand–Pettis/Weak</a></li>

<li><a href="Regulated_integral" title="Regulated integral">Regulated</a></li>

<li><a href="Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</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="Cameron%E2%80%93Martin_theorem" title="Cameron–Martin theorem">Cameron–Martin theorem</a></li>

<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a>

<ul><li><a href="Nash%E2%80%93Moser_theorem" title="Nash–Moser theorem">Nash–Moser theorem</a></li></ul></li>

<li><a href="Feldman%E2%80%93H%C3%A1jek_theorem" title="Feldman–Hájek theorem">Feldman–Hájek theorem</a></li>

<li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">No infinite-dimensional Lebesgue measure</a></li>

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

<li><a href="Structure_theorem_for_Gaussian_measures" title="Structure theorem for Gaussian measures">Structure theorem for Gaussian measures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Crinkled_arc" title="Crinkled arc">Crinkled arc</a></li>

<li><a href="Covariance_operator" title="Covariance operator">Covariance operator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_calculus" title="Functional calculus">Functional calculus</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="Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li>

<li><a href="Continuous_functional_calculus" title="Continuous functional calculus">Continuous functional calculus</a></li>

<li><a href="Holomorphic_functional_calculus" title="Holomorphic functional calculus">Holomorphic functional calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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="Banach_manifold" title="Banach manifold">Banach manifold</a> (<a href="Banach_bundle" title="Banach bundle">bundle</a>)</li>

<li><a href="Convenient_vector_space" title="Convenient vector space">Convenient vector space</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-08-15" href="https://en.wikipedia.org/wiki/?title=Bochner_measurable_function&oldid=1170505976">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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