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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:
 5.4e+41 Special maps on metric spaces
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##### References:
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