Weakly measurable function: Difference between revisions

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

<math display="block">g \circ f \colon X \to \mathbb{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>).

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>

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* {{annotated link|Bochner measurable function}}

* {{annotated link|Bochner integral}}

* {{annotated link|Bochner space}}

* {{annotated link|Pettis integral}}

* {{annotated link|Vector measure}}

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* {{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]]