×

zbMATH — the first resource for mathematics

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
Keywords:
measure space
PDF BibTeX XML Cite