Weakly measurable function: Difference between revisions

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In [[mathematics]] — specifically—specifically, in [[functional analysis]] — a—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;& \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;&rarr;&nbsp;'' \to B''</math> is said to be '''weakly measurable''' if, for every [[continuous linear functional]] ''<math>g''&nbsp; :&nbsp;'' B''&nbsp;&rarr;&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 &<math>\Sigma;</math> and the usual [[Borel sigma algebra|Borel ''&<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>(\Omega, \mathcal{P})</math> is a probability space, then a function <math>Z : \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 : B \to \mathbb{K},</math> the function

<math display="block">g \circ Z \colon \Omega \to \mathbb{K} \quad \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 <math>\Sigma</math> and the usual Borel <math>\sigma</math>-algebra on <math>\mathbb{K}.</math>

== Properties ==

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

==Properties==

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

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

<blockquote>

Function ''f'' is '''[[almost surely]] separably valued''' (or '''essentially separably valued''') if there exists a subset ''N''&nbsp;&sube;&nbsp;''X'' with ''&mu;''(''N'')&nbsp;=&nbsp;0 such that ''f''(''X''&nbsp;\&nbsp;''N'')&nbsp;&sube;&nbsp;''B'' is separable.

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

</blockquote>

A function ''<math>f''&nbsp; :&nbsp;'' X''&nbsp;&rarr;&nbsp;'' \to B''</math> defined on a [[measure space]] <math>(''X'',&nbsp;& \Sigma;,&nbsp;''& \mu;'')</math> and taking values in a Banach space ''<math>B''</math> is (strongly) measurable (withthat respectequals toa.e. &Sigma;the andlimit theof Borela ''&sigma;''-algebrasequence onof ''B''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.

== See also ==

* {{annotated link|Bochner measurable function}}

<blockquote>

* {{annotated link|Bochner integral}}

A function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''B'' defined on a [[measure space]] (''X'',&nbsp;&Sigma;,&nbsp;''&mu;'') and taking values in a Banach space ''B'' is (strongly) measurable (with respect to &Sigma; and the Borel ''&sigma;''-algebra on ''B'') [[if and only if]] it is both weakly measurable and almost surely separably valued.

* {{annotated link|Bochner space}}

</blockquote>

* {{annotated link|Pettis integral}}

* {{annotated link|Vector measure}}

== References ==

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|group=note}}

==See also==

{{reflist}}

* [[Vector-valued measure]]

* {{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}}

==References==

* {{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 analysis}}

* {{cite book

{{Analysis in topological vector spaces}}

| last = Showalter

| first = Ralph E.

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

| series = Mathematical Surveys and Monographs 49

| publisher = American Mathematical Society

| ___location = Providence, RI

| year = 1997

| pages = 103

| isbn = 0-8218-0500-2

}} {{MathSciNet|id=1422252}} (Theorem III.1.1)

[[Category:Functional analysis]]