Decomposability of Radon measures

Authors:
R. J. Gardner and W. F. Pfeffer

Journal:
Trans. Amer. Math. Soc. **283** (1984), 283-293

MSC:
Primary 28C15

DOI:
https://doi.org/10.1090/S0002-9947-1984-0735422-8

MathSciNet review:
735422

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Abstract: A topological space is called *metacompact* or *metalindelöf* if each open cover has a point-finite or point-countable refinement, respectively. It is well known that each Radon measure is expressible as a sum of Radon measures supported on a disjoint family of compact sets, called a *concassage*. If the unions of measurable subsets of the members of a concassage are also measurable, the Radon measure is called *decomposable*. We show that Radon measures in a metacompact space are always saturated, and therefore decomposable whenever they are complete. The previous statement is undecidable in ZFC if "metacompact" is replaced by "metalindelöf". The proofs are based on structure theorems for a concassage of a Radon measure. These theorems also show that the union of a concassage of a Radon measure in a metacompact space is a Borel set, which is paracompact in the subspace topology whenever the main space is regular.

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0735422-8

Keywords:
Radon measures,
Maharam measures,
decomposable measures,
metacompact and metalindelöf spaces,
weakly -refinable spaces,
continuum hypothesis,
Martin's axiom

Article copyright:
© Copyright 1984
American Mathematical Society