Let 0≤c≤1, and let η denote the order type of the set of rational numbers. Assume that with every rational number r we associate a Lebesgue-measurable subset Hr of measure c of the interval [0,1]. Prove the existence of a Lebesgue-measurable set H⊂[0,1] of measure c such that for every x∈H the set {r:x∈Hr } contains a subset of type η.
M. Laczkovich real analysisreal analysis unsolved