TITLE: Witness to a failure of Fubini/Tonelli
QUESTION [12 upvotes]: Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue measure zero?
There are such sets in models in which every set of reals of cardinality less than the continuum is Lebesgue measurable.

REPLY [6 votes]: In 1980 H. Friedman proved (in A consistent Fubini-Tonelli Theorem) that after adding $\mathfrak{c}^+$-many random reals the equality $\int f dx dy=\int f dy dx$ holds for any non-negative function for which the iterated integrals exist. In particular, a subset of the plane with the required properties does not exist in such model.