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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Cardinal conditions for strong Fubini theorems


Author: Joseph Shipman
Journal: Trans. Amer. Math. Soc. 321 (1990), 465-481
MSC: Primary 03E15; Secondary 03E35, 28A20, 28A35
MathSciNet review: 1025758
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Abstract: If $ {\kappa _1},{\kappa _2}, \ldots ,{\kappa _n}$ are cardinals with $ {\kappa _1}$ the cardinality of a nonmeasurable set, and for $ i = 2,3, \ldots ,n$ $ {\kappa _i}$ is the cardinality of a set of reals which is not the union of $ {\kappa _{i - 1}}$ measure-0 sets, then for any nonnegative function $ f:{{\mathbf{R}}^n} \to {\mathbf{R}}$ all of the iterated integrals

$\displaystyle {I_\sigma } = \iint \cdots \int {f({x_1},{x_2}, \ldots ,{x_n})d{x_{\sigma (1)}}d{x_{\sigma (2)}} \cdots d{x_{\sigma (n)}},\quad \sigma \in {S_n}} $

, which exist are equal. If all $ n!$ of the integrals exist, then the weaker condition of the case $ n = 2$ implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1025758-0
PII: S 0002-9947(1990)1025758-0
Keywords: Fubini theorem(s), real-valued measurable cardinals, nonmeasurable sets
Article copyright: © Copyright 1990 American Mathematical Society