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Math Help - Dense

  1. #1
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    Dense

    Let A be a bounded measurable subset of R, and let B be a proper measurable subset of A such that m(B)=m(A). Prove that B is dense in A.
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  2. #2
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    Suppose that B is not dense in A. Then there is a point x\in A and a ball U around x which meets no point of B. Since B is contained in A-U, m(B)\le m(A-U). Now m(A) = m(B) = m(A-U) + m(U) \ge m(B) + m(U), but m(U) must be positive, a contradiction.
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