Let (X,S,$mu$) be a measure space and M be the vector space of all real valued S-measurable functions defined on (X,S,$mu$). For $E;{in};S$ with $mu(E);<;{infty}$, $d_E$ is a pseudometric on M. With the notion of D = {$d_E$mid$E;{in};S,mu(E);<;{infty}$</TEX>}, in this paper we investigate some topological structure T of M. Indeed, we shall show that it is possible to define a complete invariant metric on M which is compatible with the topology T on M.