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 the function 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 is a probability space, then a function is called a (B-valued) weak random variable (or weak random vector) if, for every continuous linear functional the function 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 with \mu(N) = 0 such that 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.