Cardinal conditions for strong Fubini theorems

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