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.
Read more about this topic: Fubini's Theorem
Famous quotes containing the words strong and/or versions:
“Between married persons, the cement of friendship is by the laws supposed so strong as to abolish all division of possessions: and has often, in reality, the force ascribed to it.
”
—David Hume (1711–1776)
“The assumption must be that those who can see value only in tradition, or versions of it, deny man’s ability to adapt to changing circumstances.”
—Stephen Bayley (b. 1951)