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 sense. For separable spaces, the notions of weak and strong measurability agree.

Definition

If is a measurable space and B is a Banach space over a field K, then f : X → B is said to be weakly measurable if, for every continuous linear functional g : B → K, the function
is a measurable function with respect to Σ and the usual Borel σ-algebra on K.
A measurable function on a probability space is usually referred to as a random variable.
Thus, as a special case of the above definition, if is a probability space, then a function Z: : Ω → B is called a weak random variable if, for every continuous linear functional g : B → K, the function
is a K-valued random variable in the usual sense, with respect to Σ and the usual Borel σ-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 if there exists a subset N ⊆ X with μ = 0 such that f ⊆ B is separable.


Theorem . A function f : X → B defined on a measure space and taking values in a Banach space B is measurable 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.