m-topology on the ring of real-measurable functions

In this article we consider the m-topology on m(x, a ), the ring of all real measurable functions on a measurable space (x, a ), and we denote it by mm(x, a ). we show that mm(x, a ) is a hausdorff regular topological ring, moreover we prove that if (x, a ) is a t-measurable space and x is a finite set with |x| = n, then mm(x, a ) ∼= r n as topological rings. also, we show that mm(x, a ) is never a pseudocompact space and it is also never a countably compact space. we prove that (x, a ) is a pseudocompact measurable space, if and only if mm(x, a ) = mu(x, a ), if and only if mm(x, a ) is a first countable topological space, if and only if mm(x, a ) is a connected space, if and only if mm(x, a ) is a locally connected space, if and only if m∗ (x, a ) is a connected subset of mm(x, a ).