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Miklos Schweitzer 1949_4

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October 2, 2008
real analysisreal analysis unsolvedcollege contestsMiklos Schweitzer

Problem Statement

Let A A and B B be two disjoint sets in the interval (0,1) (0,1) . Denoting by μ \mu the Lebesgue measure on the real line, let μ(A)>0 \mu(A)>0 and μ(B)>0 \mu(B)>0 . Let further n n be a positive integer and \lambda \equal{}\frac1n . Show that there exists a subinterval (c,d) (c,d) of (0,1) (0,1) for which \mu(A\cap (c,d))\equal{}\lambda \mu(A) and \mu(B\cap (c,d))\equal{}\lambda \mu(B) . Show further that this is not true if λ \lambda is not of the form 1n \frac1n.
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