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Failure of Fubini's theorem for non-measurable functions

A variation of the example above shows that Fubini's theorem can fail for non-measurable functions even if | f | is integrable and both repeated integrals are well defined: if we take f to be 1 on E and –1 on the complement of E, then | f | is integrable on the product with integral 1, and both repeated integrals are well defined, but have different values 1 and –1.

Assuming the continuum hypothesis, one can identify X with the unit interval I, so there is a bounded non-negative function on I × I whose two iterated integrals (using Lebesgue measure) are both defined but unequal. This example was found by Wacław Sierpiński (1920). The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while Harvey Friedman (1980) showed that it is consistent with ZFC 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.