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 is a Banach space over a field , then is said to be weakly measurable if, for every continuous linear functional the functionis a measurable function with respect to and the usual Borel -algebra on
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 is called a weak random variable if, for every continuous linear functional the function
is a -valued random variable in the usual sense, with respect to and the usual Borel -algebra on
Properties
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.A function is said to be almost surely separably valued if there exists a subset with such that is separable.
In the case that is separable, since any subset of a separable Banach space is itself separable, one can take above to be empty, and it follows that the notions of weak and strong measurability agree when is separable.