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Two theorems on isomorphisms of measure spaces. (English. Russian original) Zbl 1445.54017
Math. Notes 104, No. 5, 758-761 (2018); translation from Mat. Zametki 104, No. 5, 781-784 (2018).
The main results of this short paper are Theorems 1 and 2:
Theorem 1. Let \((X_j,\mathcal{A}_j,\nu_j),j=1,2\), be measure spaces, where \((X_j,\mathcal{A}_j)\) are regular measurable spaces and \(\nu_j(X_j)=1\). Then the spaces \((X_1,\mathcal{A}_1,\nu_1)\) and \((X_2,\mathcal{A}_2,\nu_2)\) are isomorphic if and only if so are the spaces of atoms of the measures \(\nu_1\) and \(\nu_2\).
Theorem 2. Let \((X_1,\mathcal{A}_1,\nu_1)\) and \((X_2,\mathcal{A}_2,\nu_2)\) be regular measurable spaces with a continuous charge \(\nu_1\) and a nonatomic measure \(\nu_2\), respectively, and let \[\nu_1(X_1)=\nu_2(X_2)(>0).\] Then there exists a measurable mapping \(f\) of\(X_1\) onto \(X_2\) such that \[\nu_2=f_*(\nu_1).\]

MSC:
54E40 Special maps on metric spaces
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