Vol. 82, No. 1, 1979

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Some relationships between measures

Roy Andrew Johnson

Vol. 82 (1979), No. 1, 117–132

Suppose μ and ν are (nonnegative, countably additive) measures on the same sigma-ring. We say that ν is quasi-dominant with respect to μ if each measurable set contains a subset with the same ν-measure, where μ is absolutely continuous with respect to ν on that subset. In particular, ν is quasi-dominant with respect to μ if μ is sigma-finite. We say that ν is strongly recessive with respect to μ if the zero measure is the only measure that is quasi-dominant with respect to μ and less than or equal to ν. Properties of these relationships are investigated, and applications are given to purely atomic measures, to the Radon-Nikodým theorem and to a decomposition of product measures.

Mathematical Subject Classification 2000
Primary: 28A12
Received: 12 November 1977
Published: 1 May 1979
Roy Andrew Johnson