Fubini's nightmare

Apparent violation of Fubini's theorem

Fubini's nightmare is a seeming violation of Fubini's theorem, where a nice space, such as the square is foliated by smooth fibers, but there exists a set of positive measure whose intersection with each fiber is singular. There is no real contradiction to Fubini's theorem because despite smoothness of the fibers, the foliation is not absolutely continuous, and neither are the conditional measures on fibers.

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