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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

##### MSC:
 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
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