On some properties of the function space $\Cal M

Abstract

Let M be the vector space of all real S-measurablefunctions defined on a measure space (X, Cal S, mu ). In thispaper, we investigate some topological structure of Cal T onCal M. Indeed, (M, Cal T) becomes a topological vectorspace. Moreover, if mu is sigma-finite, we can define acomplete invariant metric on Cal M which is compatible with thetopology Cal T on Cal M, and hence (M, Cal T) becomes aF-space.