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The Lebesgue decomposition theorem for partially ordered semigroup-valued measures


Author: Panaiotis K. Pavlakos
Journal: Proc. Amer. Math. Soc. 71 (1978), 207-211
MSC: Primary 46G10; Secondary 28A55
DOI: https://doi.org/10.1090/S0002-9939-1978-0487449-4
MathSciNet review: 0487449
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Abstract: The present paper is concerned with partially ordered semigroup-valued measures. Below are given generalizations of the classical Lebesgue Decomposition Theorem. These results can be applied to Stone or ${W^ \ast }$ algebra-valued positive measures (cf. [3], [12], [13], [14]).


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Keywords: Partially ordered semigroup, monotone complete partially ordered semigroup, partially ordered semigroup of the countable type, <I>o</I>-measure, absolutely continuous and singular <I>o</I>-measure, partially ordered topological semigroup, <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\sigma$">-compatible topology with the partial ordering, <IMG WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${\tau _X}$">-measure
Article copyright: © Copyright 1978 American Mathematical Society