High Quality Content by WIKIPEDIA articles! 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. MEasurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. SPecifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.