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 \mathbb{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 \mathbb{K}, the function

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

is a measurable function with respect to \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 (\Omega, \mathcal{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 \mathbb{K}, the function

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

is a \mathbb{K}-valued random variable (i.e. measurable function) in the usual sense, with respect to \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.

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. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc.. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. MR 1501970.
  • 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. MR 1422252.