# List of Statements Undecidable in ZFC - Measure Theory

Measure Theory

A stronger version of Fubini's theorem for positive functions, 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 ZFC. On the one hand, CH implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of equivalent to a well ordering of the cardinal ω1. A similar example can be constructed using MA. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman. It can also be deduced from a variant of Freiling's axiom of symmetry.

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### Famous quotes containing the words theory and/or measure:

The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.”
George Bernard Shaw (1856–1950)

Everything is good in due measure and strong sensations know not measure.
Anton Pavlovich Chekhov (1860–1904)