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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
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A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, the proje …