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 ).