Problem 1460 (difficulty: 5/10)

Let \(\displaystyle f:\R\to\R\) be monotonically increasing and \(\displaystyle \mu_f\) the Lebesgue-Stieltjes measure generated by \(\displaystyle f\). Show that for any Borel-set \(\displaystyle H\) there are \(\displaystyle F_{\sigma}\) \(\displaystyle B\subset H\) and \(\displaystyle G_\delta\) \(\displaystyle K\supset H\) sets for which \(\displaystyle \mu_f(B)=\mu_f(K)=\mu_f(H)\).

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