Journal of Symbolic Logic

Set Theoretic Properties of Loeb Measure

Arnold W. Miller

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In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and $\kappa$-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than $\kappa$. In contrast to this we show that the additivity of Loeb measure cannot be greater than $\omega_1$. Define $\operatorname{cof}(H)$ as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that $\operatorname{card}(\lfloor\log_2(H)\rfloor) \leq \operatorname{cof}(H) \leq \operatorname{card}(2^H)$, where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes $M \preccurlyeq N$ and hyperfinite integer $H \in M$ such that $H$ is not enlarged by $N, 2^H$ contains new elements, but every new subset of $H$ has Loeb measure zero. We show that it is consistent that there exists a Sierpinski set in the reals but no Loeb-Sierpinski set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpinski sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

Article information

J. Symbolic Logic, Volume 55, Issue 3 (1990), 1022-1036.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
Secondary: 04A15 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]


Miller, Arnold W. Set Theoretic Properties of Loeb Measure. J. Symbolic Logic 55 (1990), no. 3, 1022--1036.

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