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 is a measurable space and is a Banach space over a field (which is the real numbers or complex numbers ), then is said to be weakly measurable if, for every continuous linear functional the function is 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 (or random vector if it takes values in a vector space such as the Banach space ). Thus, as a special case of the above definition, if is a probability space, then a function is called a (-valued) weak random variable (or weak random vector) if, for every continuous linear functional the function is a -valued random variable (i.e. measurable function) 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 (or essentially 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. See also References Functional analysis Measure theory Types of functions