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On a topology conforming to the convergence in measure. (English) Zbl 1099.54005
Summary: Let $$(\Omega,{\mathcal F},\mu)$$ be a measure space and let $${\mathcal S}$$ be a set consisting of all real valued functions in the wider sense and measurable functions defined on $$\Omega$$. First, we introduce a topology $${\mathcal D}$$ on $${\mathcal S}$$, next, we prove that the proposition “the sequence $$\{f_n\}$$ consisting of elements of $${\mathcal S}$$ converges to an element $$f\in{\mathcal S}$$ in the sense of measure” is equivalent to the proposition “$$f_n$$ converges to $$f$$ in the sense of the topology $${\mathcal D}$$”, and last we show that the topological space $$({\mathcal S},{\mathcal D})$$ becomes a Hausdorff space and satisfies the first countability axiom.
##### MSC:
 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54C30 Real-valued functions in general topology 54C35 Function spaces in general topology 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
measure space