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Math Help - Measurability of a Product Space

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    Measurability of a Product Space

    We know that if A in R^n and B in R^m are (lebesgue) measurable sets, then so is the set AxB in R^(n+m).

    Prove that the converse is also true. That is, if AxB is measurable, then so is A and B.
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    Quote Originally Posted by southprkfan1 View Post
    We know that if A in R^n and B in R^m are (lebesgue) measurable sets, then so is the set AxB in R^(n+m).

    Prove that the converse is also true. That is, if AxB is measurable, then so is A and B.
    As stated, that result is false. For example, if S is a non-measurable subset of \mathbb{R} then S\times\{0\} is a null set (hence measurable) in \mathbb{R}^2.

    To make the result correct, you need to add the hypothesis that A\times B has measure greater than 0. Then (from the arguments used to prove Fubini's theorem) almost all the slices of A\times B are measurable. But in a product set all the slices are equal to A (for horizontal slices) or B (for vertical slices). Therefore A and B are measurable.
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