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I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.

Let $S$ be a subset of a region, say $R$ which is enclosed by a rectangle. Assume that if any line, say $L$ parallel to each side (both horizontal and vertical lines) of the rectangle, then $L \cap S$ has full (Lebesgue) measure on $L$.

  1. If $S$ is measurable, does $S$ have the full measure on $R$?
  2. If $S$ is non-measurable, does $S$ have to contain a (proper) subset which has the full measure on $R$ with respect to the product (Lebesgue) measure? If yes, how could I show it?

Thank you for your answer or any book or references.

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  • $\begingroup$ See mathoverflow.net/q/160329/22277. $\endgroup$
    – Joseph Van Name
    Commented Jul 12, 2014 at 3:06
  • $\begingroup$ I don't see how the answer to the linked question addresses question 1 above. $\endgroup$
    – Vincent
    Commented Aug 6, 2014 at 10:26
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    $\begingroup$ Question 1 is answered by Fubini’s theorem. $\endgroup$
    – Emil Jeřábek
    Commented Aug 6, 2014 at 12:47
  • $\begingroup$ I modified this question yesterday. The previous question is just whether S has full measure or not. However, I found that the above second question is actually what I would like to know. $\endgroup$
    – Young Woo Nam
    Commented Aug 8, 2014 at 1:22

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