Weakly measurable function: Difference between revisions

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In [[mathematics]]—specifically, in [[functional analysis]]—a '''weakly measurable function''' taking values in a [[Banach space]] is a [[functionFunction (mathematics)|function]] whose [[functionFunction composition|composition]] with any element of the [[continuousContinuous dual space|dual space]] is a [[measurable function]] in the usual (strong) sense. For [[separable space]]s, the notions of weak and strong measurability agree.

== Definition ==

If <math>(''X'',&nbsp;Σ \Sigma)</math> is a [[measurable space]] and ''<math>B''</math> is a Banach space over a [[fieldField (mathematics)|field]] '''<math>\mathbb{K'''}</math> (usuallywhich is the [[real number]]s '''<math>\R'''</math> or [[complex number]]s '''C'''<math>\Complex</math>), then ''<math>f''&nbsp; :&nbsp;'' X''&nbsp;→&nbsp;'' \to B''</math> is said to be '''weakly measurable''' if, for every [[continuous linear functional]] ''<math>g''&nbsp; :&nbsp;'' B''&nbsp;→&nbsp;''' \to \mathbb{K'''},</math> the function

:<math display="block">g \circ f \colon X \to \mathbfmathbb{K} \colonquad \text{ defined by } \quad x \mapsto g(f(x))</math>

is a measurable function with respect to Σ<math>\Sigma</math> and the usual [[Borel sigma algebra|Borel ''σ''<math>\sigma</math>-algebra]] on '''<math>\mathbb{K'''}.</math>

A measurable function on a [[probability space]] is usually referred to as a [[random variable]] (or [[random vector]] if it takes values in a vector space such as the Banach space ''<math>B''</math>).

:<math>g \circ f \colon X \to \mathbf{K} \colon x \mapsto g(f(x))</math>

Thus, as a special case of the above definition, if <math>(Ω\Omega,&nbsp;Σ,&nbsp;''' \mathcal{P'''})</math> is a probability space, then a function ''<math>Z'' :&nbsp;:&nbsp;Ω&nbsp;→&nbsp;'' \Omega \to B''</math> is called a (''<math>B''</math>-valued) '''weak random variable''' (or '''weak random vector''') if, for every continuous linear functional ''<math>g''&nbsp; :&nbsp;'' B''&nbsp;→&nbsp;''' \to \mathbb{K'''},</math> the function

:<math display="block">g \circ Z \colon \Omega \to \mathbfmathbb{K} \colonquad \text{ defined by } \quad \omega \mapsto g(Z(\omega))</math>

is a '''<math>\mathbb{K'''}</math>-valued random variable (i.e. measurable function) in the usual sense, with respect to Σ<math>\Sigma</math> and the usual Borel ''σ''<math>\sigma</math>-algebra on '''<math>\mathbb{K'''}.</math>

== Properties ==

is a measurable function with respect to Σ and the usual [[Borel sigma algebra|Borel ''σ''-algebra]] on '''K'''.

The relationship between measurability and weak measurability is given by the following result, known as '''[[B. J. Pettis|Pettis]]' theorem''' or '''Pettis measurability theorem'''.

A measurable function on a [[probability space]] is usually referred to as a [[random variable]] (or [[random vector]] if it takes values in a vector space such as the Banach space ''B'').

Thus, as a special case of the above definition, if (Ω,&nbsp;Σ,&nbsp;'''P''') is a probability space, then a function ''Z'':&nbsp;:&nbsp;Ω&nbsp;→&nbsp;''B'' is called a (''B''-valued) '''weak random variable''' (or '''weak random vector''') if, for every continuous linear functional ''g''&nbsp;:&nbsp;''B''&nbsp;→&nbsp;'''K''', the function

A function ''<math>f''</math> is said to be '''[[almost surely]] separably valued''' (or '''essentially separably valued''') if there exists a subset ''<math>N''&nbsp;⊆&nbsp;'' \subseteq X''</math> with ''μ''<math>\mu(''N'')&nbsp; =&nbsp; 0</math> such that ''<math>f''(''X''&nbsp; \&nbsp;''setminus N'')&nbsp;⊆&nbsp;'' \subseteq B''</math> is separable.

:<math>g \circ Z \colon \Omega \to \mathbf{K} \colon \omega \mapsto g(Z(\omega))</math>

{{math theorem|name=Theorem|note=Pettis, 1938|style=|math_statement=

is a '''K'''-valued random variable (i.e. measurable function) in the usual sense, with respect to Σ and the usual Borel ''σ''-algebra on '''K'''.

'''Theorem''' (Pettis, 1938)'''.''' A function ''<math>f''&nbsp; :&nbsp;'' X''&nbsp;→&nbsp;'' \to B''</math> defined on a [[measure space]] <math>(''X'',&nbsp;Σ \Sigma,&nbsp;''μ'' \mu)</math> and taking values in a Banach space ''<math>B''</math> is (strongly) measurable (that equals a.e. the limit of a sequence of measurable countably-valued functions) [[if and only if]] it is both weakly measurable and almost surely separably valued.

}}

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

==Properties==

== See also ==

The relationship between measurability and weak measurability is given by the following result, known as '''[[B. J. Pettis|Pettis]]' theorem''' or '''Pettis measurability theorem'''.

* [[{{annotated link|Bochner measurable function]]}}

<blockquote>

* [[{{annotated link|Bochner integral]]}}

A function ''f'' is said to be '''[[almost surely]] separably valued''' (or '''essentially separably valued''') if there exists a subset ''N''&nbsp;⊆&nbsp;''X'' with ''μ''(''N'')&nbsp;=&nbsp;0 such that ''f''(''X''&nbsp;\&nbsp;''N'')&nbsp;⊆&nbsp;''B'' is separable.

* {{annotated link|Bochner space}}

</blockquote>

* [[{{annotated link|Pettis integral]]}}

* {{annotated link|Vector measure}}

== References ==

<blockquote>

'''Theorem''' (Pettis, 1938)'''.''' A function ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''B'' defined on a [[measure space]] (''X'',&nbsp;Σ,&nbsp;''μ'') and taking values in a Banach space ''B'' is (strongly) measurable (that equals a.e. the limit of a sequence of measurable countably-valued functions) [[if and only if]] it is both weakly measurable and almost surely separably valued.

</blockquote>

{{reflist|group=note}}

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

{{reflist}}

* {{cite journal|last=Pettis|first=B. J.|authorlink=Billy James Pettis|title=On integration in vector spaces|journal=Trans. Amer. Math. Soc.|volume=44|year=1938|number=2|pages=277&ndash;304|issn=0002-9947|mr=1501970|doi=10.2307/1989973|doi-access=free}}

==See also==

* {{cite book|last=Showalter|first=Ralph E.|title=Monotone operators in Banach space and nonlinear partial differential equations|url=https://archive.org/details/monotoneoperatio00show|url-access=limited|series=Mathematical Surveys and Monographs 49|publisher=American Mathematical Society|___location=Providence, RI|year=1997|page=[https://archive.org/details/monotoneoperatio00show/page/n109 103]|isbn=0-8218-0500-2|mr=1422252|contribution=Theorem III.1.1}}

* [[Bochner measurable function]]

* [[Bochner integral]]

* [[Pettis integral]]

* [[Vector-valued measure]]

== References ==

* {{cite journal

| last = Pettis

| first = B. J.

| authorlink = Billy James Pettis

| title = On integration in vector spaces

| journal = Trans. Amer. Math. Soc.

| volume = 44

| year = 1938

| number = 2

| pages = 277&ndash;304

| issn = 0002-9947

| mr = 1501970

| doi = 10.2307/1989973| doi-access = free

}}

* {{cite book

| last = Showalter

| first = Ralph E.

| title = Monotone operators in Banach space and nonlinear partial differential equations

| url = https://archive.org/details/monotoneoperatio00show

| url-access = limited

| series = Mathematical Surveys and Monographs 49

| publisher = American Mathematical Society

| ___location = Providence, RI

| year = 1997

| page = [https://archive.org/details/monotoneoperatio00show/page/n109 103]

| isbn = 0-8218-0500-2

| mr=1422252

| contribution = Theorem III.1.1}}

{{Functional Analysisanalysis}}

{{Analysis in topological vector spaces}}

{{AnalysisInTopologicalVectorSpaces}}

[[Category:Functional analysis]]