weakly measurable function

If $(X,\; \backslash Sigma)$ is a measurable space and $B$ is a Banach space over a field $\backslash mathbb\{K\}$ (which is the real numbers $\backslash R$ or complex numbers $\backslash Complex$), then $f\; :\; X\; \backslash to\; B$ is said to be weakly measurable if, for every continuous linear functional $g\; :\; B\; \backslash to\; \backslash mathbb\{K\},$ the function

$$g\; \backslash circ\; f\; \backslash colon\; X\; \backslash to\; \backslash mathbb\{K\}\; \backslash quad\; \backslash text\{\; defined\; by\; \}\; \backslash quad\; x\; \backslash mapsto\; g(f(x))$$

is a measurable function with respect to $\backslash Sigma$ and the usual Borel $\backslash sigma$-algebra on $\backslash mathbb\{K\}.$

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 $(\backslash Omega,\; \backslash mathcal\{P\})$ is a probability space, then a function $Z\; :\; \backslash Omega\; \backslash to\; B$ is called a ($B$-valued) weak random variable (or weak random vector) if, for every continuous linear functional $g\; :\; B\; \backslash to\; \backslash mathbb\{K\},$ the function

$$g\; \backslash circ\; Z\; \backslash colon\; \backslash Omega\; \backslash to\; \backslash mathbb\{K\}\; \backslash quad\; \backslash text\{\; defined\; by\; \}\; \backslash quad\; \backslash omega\; \backslash mapsto\; g(Z(\backslash omega))$$

is a $\backslash mathbb\{K\}$-valued random variable (i.e. measurable function) in the usual sense, with respect to $\backslash Sigma$ and the usual Borel $\backslash sigma$-algebra on $\backslash mathbb\{K\}.$

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

A function $f$ is said to be almost surely separably valued (or essentially separably valued) if there exists a subset $N\; \backslash subseteq\; X$ with $\backslash mu(N)\; =\; 0$ such that $f(X\; \backslash setminus\; N)\; \backslash subseteq\; B$ is separable.

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

A function $f\; :\; X\; \backslash to\; B$ defined on a measure space $(X,\; \backslash Sigma,\; \backslash mu)$ 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.

}}

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.

• {{annotated link|Bochner measurable function}}
• {{annotated link|Bochner integral}}
• {{annotated link|Bochner space}}
• {{annotated link|Pettis integral}}
• {{annotated link|Vector measure}}

{{reflist|group=note}}

{{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=277–304|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 analysis}}

{{Analysis in topological vector spaces}}

Category:Functional analysis

Category:Measure theory

Category:Types of functions