Strong Versions
The stronger versions of Fubini's theorem, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of the standard Zermelo–Fraenkel axioms of set theory. Martin's axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while a variant of Freiling's axiom of symmetry implies that a strong Fubini-type theorem for does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.
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