SOME ALGEBRAIC AND TOPOLOGICAL PROPERTIES OF RINGS OF MEASURABLE FUNCTIONS

For the ring M(X, A) of all real valued measurable functions over a measurable space (X, A), Mc(X, A) stands for the subring of M(X, A) consisting of those functions that have countable range in R. The structure space of Mc(X, A) is shown to be homeomorphic to that of M(X, A) and we observe that within the class of separated realcompact measurable spaces, M(X, A) is isomorphic to M(Y, 13) when and only when Mc(X, A) is isomor-phic to Mc(Y, 13). We also introduce, via a measure mu on (X, A), three differ-ent topologies on M(X, A), namely the U mu-topology, U mu F-topology and mF mu- topology. We establish a necessary and sufficient criterion for mF nu-topology to be weaker than the mF mu-topology. We also give a necessary and suffi-cient condition for the ring M(X, A) to be a topological ring (equivalently, a topological vector space) in the U mu F-topology. We further observe that this condition holds if and only if the mF mu-topology on M(X, A) is connected if and only if the pseudonorm topology on LF (X, A, mu), the set of all weakly essentially bounded functions in M(X, A), coincides with its relative mF mu- topology inherited from M(X, A). Additionally, we show that a maximal ideal M in M(X, A) is closed in the U-topology if and only if it is real. Two special subrings of M(X, A), namely MF (X, A, mu) and M & INFIN;(X, A, mu), and the associated ideal structures of the parent ring M(X, A) are also studied.