Problem 1507 (difficulty: 5/10)

Let \(\displaystyle f:C\to[0,1]\) be the Cantor-function. For each \(\displaystyle H\subset[0,1]\) Borel-set let \(\displaystyle \mu_1(H)=\lambda(f(H\cap C))\), \(\displaystyle \mu_2(H)=\lambda(f^{-1}(H))\) and \(\displaystyle \mu_3=\mu_1+\mu_2\). Which pairs of the measures \(\displaystyle \mu_1\), \(\displaystyle \mu_2\), \(\displaystyle \mu_3\) and \(\displaystyle \lambda\) are singular, absolutely continuous? What is the Lebesgue decomposition of the measures \(\displaystyle \mu_i\) with respect to Lebesgue-measure? What is the Lebesgue decomposition of Lebesgue-measure with respect to the \(\displaystyle \mu_i\)?

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