an:05034014
Zbl 1099.54005
Niizeki, S.; Kuwano, H.; Araki, M.
On a topology conforming to the convergence in measure
EN
Kochi J. Math. 1, 67-75 (2006).
00124398
2006
j
54A20 54C30 54C35 28A20
measure space
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.