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