Slicing up compact sets

Show that if K is a compact subset of \Bbb R^n, and if every intersection of K with an hyperplane x_n=a has measure zero in \Bbb R^{n-1}, then K has measure zero.

Note This is valid for any measurable set in any \sigma-compact measure space by virtue of Tonelli’s theorem. The point is to prove this without using this. The fact that K is compact will come in handy.

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Slicing up compact sets

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