Weakly measurable function

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.

A function f : X \to B defined on a measure space (X, \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.

References