Weakly measurable function explained

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, \Sigma) is a measurable space and B is a Banach space over a field K (which is the real numbers \R or complex numbers \Complex ), then f : X \to B is said to be weakly measurable if, for every continuous linear functional g : B \to K , the function$$g\; \backslash circ\; f\; \backslash colon\; X\; \backslash to\; \backslash mathbb\; \backslash quad\; \backslash text\; \backslash quad\; x\; \backslash mapsto\; g(f(x))$$is a measurable function with respect to \Sigma and the usual Borel \sigma -algebra on 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 (\Omega, l{P}) is a probability space, then a function Z : \Omega \to B is called a (B -valued) weak random variable (or weak random vector) if, for every continuous linear functional g : B \to K , the function$$g\; \backslash circ\; Z\; \backslash colon\; \backslash Omega\; \backslash to\; \backslash mathbb\; \backslash quad\; \backslash text\; \backslash quad\; \backslash omega\; \backslash mapsto\; g(Z(\backslash omega))$$is a K -valued random variable (i.e. measurable function) in the usual sense, with respect to \Sigma and the usual Borel \sigma -algebra on 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.

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.

References

• Pettis. B. J.. Billy James Pettis. On integration in vector spaces. Trans. Amer. Math. Soc.. 44. 1938. 2. 277 - 304. 0002-9947. 1501970. 10.2307/1989973. free.
• Book: Showalter, Ralph E.. Monotone operators in Banach space and nonlinear partial differential equations. limited. Mathematical Surveys and Monographs 49. American Mathematical Society. Providence, RI. 1997. 103. 0-8218-0500-2. 1422252. Theorem III.1.1.