2025/05/26 by Amrita Dey, Dey, Amrita · 1 voice
Mathematics · #Advanced Banach Space Theory #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.GN
paper · pdf · doi:10.48550/arxiv.2505.19780
openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a probability measure space (X,\mathscrA,μ), the topology Mμ, is defined on the ring M(X,\mathscrA,μ) of real-valued measurable functions on (X,\mathscrA,μ) involving the notion of convergence in measure. It turns out that if f=g is assumed to be in the almost everywhere sense, Mμ is a completely metrizable space and is induced by the metric given by δ(f,g)=μ(X∖ Z(f-g)), for f,g∈ M(X,\mathscrA,μ). The notion of a measure being bounded away from zero is introduced and it is observed that a measure μ is bounded away from zero if and only if it is purely atomic and contains at most finitely many pairwise disjoint atoms. Topological properties, such as being a P-space, extremal disconnectedness and local connectedness of Mμ are found to be equivalent to the underlying measure μ being bounded away from zero. The space Mμ is proven to be never Lindelöf, and hence cannot be separable, second countable or compact. It is established that Mμ is connected (in fact, path-connected) if and only if μ is non-atomic and Mμ is totally disconnected if and only if μ is purely atomic. The component of a point in Mμ (which is found to be equivalent to the path-component and quasicomponent of that point in Mμ) is computed in a general setting.