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Real-valued-measurable cardinals. (English) Zbl 0839.03038
Judah, Haim (ed.), Set theory of the reals. Proceedings of a winter institute on set theory of the reals held at Bar-Ilan University, Ramat-Gan (Israel), January 1991. Providence, RI: American Mathematical Society (Distrib.), Isr. Math. Conf. Proc. 6, 151-304 (1993).
The author begins his introduction with the words: “In these notes I seek to describe the theory of real-valued-measurable cardinals, collecting together various results which are scattered around the published literature, and including a good deal of unpublished material”.
A real-valued-measurable cardinal is a cardinal \(\kappa\) for which there is a \(\kappa\)-additive measure \(\mu: {\mathcal P}\kappa\to [0,1 ]\) with \(\mu (\kappa)=1\) and with \(\mu\) zero on singleton sets. An atom for the measure \(\mu\) is a subset \(E\) of \(\kappa\) with \(\mu(E)>0\) and either \(\mu(F) =0\) or \(\mu (E\setminus F)=0\) for all \(F\subseteq E\). The measure is two-valued if \(\mu(X)\in \{0,1\}\) for all \(X\subseteq \kappa\). A classical result of S. Ulam [Fundam. Math. 16, 140-150 (1930; JFM 56.0920.04)] is that a cardinal is real-valued-measurable if and only if it is either atomlessly-measurable or two-valued-measurable, and further there is an extension of Lebesgue measure to a measure defined on every subset of the reals if and only if there is an atomlessly-measurable cardinal. This survey concentrates on atomlessly-measurable cardinals. As the author says: “The investigation will take us into a fascinating blend of measure theory, infinitary combinatorics and metamathematics, drawing on deep ideas from all three”.
There are nine sections in the paper. Section 1 is devoted to basic material (including Ulam’s results mentioned above). Section 2 discusses various consistency results including Solovay’s theorem that the theory ZFC + (there is an atomlessly-measurable cardinal) is equiconsistent with the theory ZFC + (there is a two-valued-measurable cardinal). Section 3 considers results of M. Gitik and S. Shelah [Isr. J. Math. 68, No. 2, 129-160 (1989; Zbl 0686.03027)] showing the complexity of the measure algebra from an atomlessly-measurable cardinal. Section 4, whimsically entitled “The enormity of real-valued-measurable cardinals”, has results showing the large size of a real-valued-measurable cardinal; in particular that there are many \(\Pi^2_0\)-indescribable cardinals below a real-valued-measurable cardinal. Section 5 looks at combinatorial consequences of supposing \(\kappa\) to be an atomlessly-measurable cardinal, section 6 looks at measure-theoretic implications, and section 7 has some results on partially ordered sets. Section 8 describes two assertions, both stronger than the assertion that \(2^{\aleph_0}\) is real-valued-measurable, with some of their consequences in set theory and general topology. Section 9 mentions some extensions outside ordinary measure theory of the earlier ideas, to quasi-measurable cardinals. There is then some 40 pages of appendix, giving definitions and theorems in combinatorics, measure theory, general topology and indescribable cardinals that have found mention in the main body of the paper. The paper concludes with a discussion of some open problems.
For the entire collection see [Zbl 0821.00016].

03E35 Consistency and independence results
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
28E15 Other connections with logic and set theory
03E55 Large cardinals
28A12 Contents, measures, outer measures, capacities