# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] K. Kuratowski, Topology, Vol. 1 (Academic Press, New York-London; Panstwowe Wydawnictwo Naukowe, Warsaw, 1966; Mir, Moscow, 1966). · Zbl 0158.40901 [2] V. I. Bogachev and O.G. Smolyanov, Real and Functional Analysis (NITs “Regular and Chaotic Dynamics,” Moscow-Izhevsk, 2011) [in Russian]. [3] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1972; “Introductory Real Analysis,” Dover Publications, Inc., New York, 1975). [4] G. E. Shilov, Mathematical Analysis. Second Special Course (Nauka, Moscow, 1965) [in Russian]. [5] A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Functional Analysis (Nauka,Moscow, 1979; Springer-Verlag, New York-Berlin, 1982). · Zbl 0486.46002 [6] V. I. Bogachev, Foundations of Measure Theory, Vol. 2 (NITs “Regular and Chaotic Dynamics”, Moscow-Izhevsk, 2003; “Measure Theory,” Vol. II. Springer-Verlag, Berlin, 2007). [7] Moser, J., No article title, Trans. Amer. Math. Soc., 120, 286, (1965) [8] Greene, R. E.; Shiohama, K., No article title, Trans. Amer. Math. Soc., 255, 403, (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.