-finiteness and -additivity of measures on sets and left invariant measures on discrete groups

Authors:
Thomas E. Armstrong and Karel Prikry

Journal:
Proc. Amer. Math. Soc. **80** (1980), 105-112

MSC:
Primary 28C10; Secondary 03E55, 28A12

MathSciNet review:
574517

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Abstract: For any cardinal a possibly infinite measure on a set *X* is strongly non--additive if *X* is partitioned into or fewer -negligible sets. The measure is purely non--additive if it dominates no nontrivial -additive measure. The properties and relationships of these types of measures are examined in relationship to measurable ideal cardinals and real-valued measurable cardinals. Any -finite left invariant measure on a group *G* of cardinality larger than is strongly non--additive. In particular, -finite left invariant measures on infinite groups are strongly finitely additive.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0574517-0

Keywords:
Left invariant measures,
left invariant means,
-additivity,
-finiteness,
pure non--additivity,
real-valued measurable cardinal,
-complete ideal,
-saturated ideal

Article copyright:
© Copyright 1980
American Mathematical Society