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.
Read more about this topic: List Of Statements Undecidable In ZFC
Famous quotes containing the words measure and/or theory:
“Letters are above all useful as a means of expressing the ideal self; and no other method of communication is quite so good for this purpose.... In letters we can reform without practice, beg without humiliation, snip and shape embarrassing experiences to the measure of our own desires....”
—Elizabeth Hardwick (b. 1916)
“Won’t this whole instinct matter bear revision?
Won’t almost any theory bear revision?
To err is human, not to, animal.”
—Robert Frost (1874–1963)