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Cardinal conditions for strong Fubini theorems. (English) Zbl 0715.03022
W. Sierpinski observed in 1920 that the continuum hypothesis CH implies that there exists a subset of [0,1]\(\times [0,1]\) that is countable on every horizontal line and co-countable on every vertical line. (Take a well-ordering of [0,1] of order-type \(\aleph_ 1.)\)
As CH is consistent with the ZFC system of set-theoretical axioms, one cannot prove in ZFC a strong Fubini theorem implying that iterated integrals of bounded functions never disagree. The author introduces, for each natural number n, the following condition \((*_ n):\) there exist cardinal numbers \(\kappa_ 1,\kappa_ 2,...,\kappa_ n\) such that \(\kappa_ 1\) is the cardinality of a non-Lebesgue-measurable set of real numbers and for \(i=2,...,n\), \(\kappa_ i\) is the cardinality of a set of reals which is not the union of \(\kappa_{i-1}\) measure-0-sets. He also defines, for each natural number n the statement \((B_ n):\) If f: \({\mathbb{R}}^ n\to {\mathbb{R}}\) is nonnegative and any two of the integrals \(I_{\sigma}\), \(\sigma \in S_ n\) (with \[ I_{\sigma}=\iint...\int f(x_ 1,x_ 2,...,x_ n)dx_{\sigma (1)}dx_{\sigma (2)}...dx_{\sigma (n)}) \] exist, then those two integrals are equal.
Quoting a paper by K. Kunen from 1984, he observes that the \((*)_ n\) are all consistent with ZFC.
His two main results are: \((*)_ n\) implies \(B_ n\), and: the existence of a real-valued measurable cardinal implies \((*)_ n\). In the proof of the first result he uses, among other things, an argument of C. Freiling from 1986.
The paper contains several interesting observations, suggestions and open problems; it is clearly written and a pleasure to read.
Reviewer: W.Veldman

03E55 Large cardinals
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
03E35 Consistency and independence results
03E65 Other set-theoretic hypotheses and axioms
Full Text: DOI
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