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Set theoretic properties of Loeb measure. (English) Zbl 0721.03050

Let M be an \(\omega_ 1\)-saturated model of PA, and H an M-finite set. The Loeb measure is a natural, countably additive measure on the \(\sigma\)-algebra generated by the subsets of H coded in M. The theorems in this paper are a smattering of results about the Loeb measure, often under additional assumptions about M or about V (the ambient model of set theory in which M lives). For instance, it is shown that H cannot be covered by fewer than \(\kappa\)-many sets of Loeb measure 0, if \(MA_{\kappa}\) holds in V and if M is \(\kappa\)-saturated. (By contrast, it is also shown that the Loeb measure is never \(\omega_ 2\)-additive, with no special hypotheses.) For another example, let cof(H) be the smallest cardinality of a family F of Loeb measure 0 sets such that each set of measure 0 is a subset of some member of F. The author proves that \(card(\lfloor \log_ 2(H)\rfloor)\leq cof(H)\leq card(2^ H)\) (where card is the external cardinality function). Some other theorems are relative consistency results, as in the following examples. A Sierpiński set is an uncountable set of reals which has only a countable intersection with each measure 0 set. A Loeb-Sierpiński set is the analogous notion for the Loeb measure: an uncountable subset of H which has only a countable intersection with each Loeb measure 0 set. The author shows that it is consistent with ZFC that there is a Sierpiński set in \(2^{\omega}\) but no Loeb-Sierpiński set for any H in any M; he also shows that it is consistent with ZFC that CH fails and that there is a Loeb-Sierpiński set for some H in some M. There are also theorems, with no additional hypotheses, about elementary extensions N of M and Loeb measure 0 sets; namely, there is such an N in which \(cof(H)=\omega_ 1\), and there is such an N with new subsets of H but no new measure 0 subsets of H.
Reviewer’s remark: The interested reader should be advised that some of the results stated in the author’s abstract are actually the opposites of what are proven!

MSC:

03H05 Nonstandard models in mathematics
28E05 Nonstandard measure theory
03E35 Consistency and independence results
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References:

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