The complementation of an additive measure up to $\sigma$-additivity by means of an extension of the space

D. N. Dudin

M. V. Lomonosov Moscow State University

Abstract: For an algebra $\mathfrak A$ of subsets of a set X there is constructed a set $\widetilde X\supset X$ and an algebra of its subsets so that the mapping $\widetilde A\to A=\mathop\mathfrak A\limits^\sim\cap A$ is a one-to-one correspondence between $\mathop\mathfrak A\limits^\sim$ and $\mathfrak A$ and for each additive measure $M$ on $\mathfrak A$ the measure $\widetilde\mu$ on $\mathop\mathfrak A\limits^\sim$ defined by the equation $\widetilde\mu(\widetilde A)=\mu(A)$ is countably additive.

UDC: 517.5

Received: 14.06.1967

 English version: Mathematical Notes, 1968, 3:1, 42–44

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