A generalization of m-topology and U-topology on rings of measurable functions

In this paper, we generalize mμ and Uμ topologies on M ( X , A ) via an ideal I in the ring M ( X , A ) of all real-valued measurable functions. The collection LI∞ ( μ ) of all essentially I-bounded functions over the measure space (X,A,μ)$ and the ideal Iμ (X,A) ={ f ∈ M (X,A) : for every g ∈ M(X,A), fg is essentially I-bounded } are the components of 0 in the space mμI and UμI respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide. In particular, it is proved that they coincide if and only if the three sets M(X,A), LI∞ ( μ ) , and Iμ (X,A) coincide and this holds when and only when M(X,A), with either topology, is connected, which is further equivalent to each of these two topological spaces being a topological ring as well as a topological vector space. Furthermore, LI∞ ( μ ) is a complete pseudonormed linear space regardless of the choice of I. Finally, we examine when these later spaces are separable.