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I am starting study on Lebesgue-Stieltjes measure. One of the problem that stuck me is:

If $\mu$ is a Radon measure on $R$, show that there exists a monotone function $F: R \to R$ such that $\mu = \mu_F$. (Here $\mu_F$ is the Lebesgue-Stieletjes measure of $F$).

I'm thinking of construct $F(x) = \mu (-\infty, x)$, but this doesn't work since $F(b) -F(a)$ is always finite but $\mu(-\infty,a)$ and $\mu(-\infty,b)$ can both be infinite.

Any idea is appreciated.

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