×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chris Freiling, Axioms of symmetry: throwing darts at the real number line, J. Symbolic Logic 51 (1986), no. 1, 190 – 200. · Zbl 0619.03035 · doi:10.2307/2273955 · doi.org
[2] Fremlin, [1983] Cichón’s diagram, Séminaire Initiation à l’Analyse 23 (1983/84), no. 5.
[3] D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. (Rozprawy Mat.) 260 (1987), 116. · Zbl 0703.28003
[4] Harvey Friedman, A consistent Fubini-Tonelli theorem for nonmeasurable functions, Illinois J. Math. 24 (1980), no. 3, 390 – 395. · Zbl 0467.28003
[5] Moti Gitik and Saharon Shelah, Forcings with ideals and simple forcing notions, Israel J. Math. 68 (1989), no. 2, 129 – 160. · Zbl 0686.03027 · doi:10.1007/BF02772658 · doi.org
[6] Philosophy of mathematics: Selected readings, Edited and with an introduction by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[7] Stanley P. Gudder, Probability manifolds, J. Math. Phys. 25 (1984), no. 8, 2397 – 2401. · doi:10.1063/1.526461 · doi.org
[8] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. · Zbl 0419.03028
[9] Kenneth Kunen, Random and Cohen reals, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 887 – 911. · Zbl 0588.03035
[10] Itamar Pitowsky, Deterministic model of spin and statistics, Phys. Rev. D (3) 27 (1983), no. 10, 2316 – 2326. · doi:10.1103/PhysRevD.27.2316 · doi.org
[11] A. N. Shiryayev, Probability, Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas.
[12] Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1 – 56. · Zbl 0207.00905 · doi:10.2307/1970696 · doi.org
[13] Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397 – 428.
[14] Ulam, [1930] Zur Masstheorie in der algemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. · JFM 56.0920.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.