Cardinal conditions for strong Fubini theorems.

*(English)*Zbl 0715.03022W. 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.

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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\textit{J. Shipman}, Trans. Am. Math. Soc. 321, No. 2, 465--481 (1990; Zbl 0715.03022)

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##### 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 |

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