Weakly measurable function

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

Definition

If $(X,\mathrm{\Sigma })$ is a measurable space and $B$ is a Banach space over a field $\mathbb{𝕂}$ (which is the real numbers $\mathbb{ℝ}$ or complex numbers $\mathbb{ℂ}$), then $f:X\to B$ is said to be weakly measurable if, for every continuous linear functional $g:B\to \mathbb{𝕂},$ the function $$g\circ f:X\to \mathbb{𝕂}\text{ defined by }x\mapsto g(f(x))$$ is a measurable function with respect to $\mathrm{\Sigma }$ and the usual Borel $\sigma$-algebra on $\mathbb{𝕂}.$

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 $(\mathrm{\Omega },{𝒫})$ is a probability space, then a function $Z:\mathrm{\Omega }\to B$ is called a ($B$-valued) weak random variable (or weak random vector) if, for every continuous linear functional $g:B\to \mathbb{𝕂},$ the function $$g\circ Z:\mathrm{\Omega }\to \mathbb{𝕂}\text{ defined by }\omega \mapsto g(Z(\omega ))$$ is a $\mathbb{𝕂}$-valued random variable (i.e. measurable function) in the usual sense, with respect to $\mathrm{\Sigma }$ and the usual Borel $\sigma$-algebra on $\mathbb{𝕂}.$

Properties

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 $N\subseteq X$ with $\mu (N)=0$ such that $f(X\setminus N)\subseteq B$ is separable.

Theorem (Pettis, 1938) — A function $f:X\to B$ defined on a measure space $(X,\mathrm{\Sigma },\mu )$ 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.

See also

• Bochner measurable function
• Bochner integral
• Bochner space – Mathematical concept
• Pettis integral
• Vector measure

References

• Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. 
• Showalter, Ralph E. (1997). "Theorem III.1.1". Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 103. ISBN 0-8218-0500-2. https://archive.org/details/monotoneoperatio00show.