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

is a measurable space is a Banach space over a field (which is the real numbers or complex numbers is said to be weakly measurable if, for every continuous linear functional and the usual \sigma-algebra

A measurable function on a probability space is usually referred to as a random variable (or random vector-valued) weak random variable (or weak random vector

Properties

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.

is said to be almost surely separably valued (or essentially separably valued is separable.

is separable.

See also

References