I got to have this problem in my hand.

Problem Let $F: \mathbb{R}\rightarrow \mathbb{R}$ be an increasing, right continuous function, and let $\phi :\mathbb{R}\rightarrow \mathbb{R}$ be a continuous increasing invertible function. Let $\mu_F$ and $\mu_{F\circ \phi}$ be the Lebesgue-Stieljes measure associated to $F$ and $F\circ \phi$ respectively. Show that if $f\in L^1(\mu_F)$, then $f\circ \phi \in L^1(\mu_{F\circ \phi})$ and $$ \int f d\mu_F = \int f\circ \phi d\mu_{F\circ \phi}$$ Hint: It is enough to consider non-negative $f$ and to prove the inequality $\int f\circ \phi d\mu_{F\circ \phi}\leq \int f\mu_F$.

There were many situation I have studied this kind of problem and I was thinking of using simple function and generalize it to arbitrary $L^1$ functions. And I think the method just works here too. But I can not understand the hint on this problem. Because I started from simple, the hint for non negative function seems superfluous but makes sense. But I don't know how to use the hint for inequality. I will be happy with any kind of hint or comment on it. Thank you in advance!

  • 1
    $\begingroup$ If you prove the hint for positive functions, you show that $f\circ \phi \in L^1(\mu_{F\circ \phi})$, since you are taking absolute values in any case. The change of variables formula is, indeed, proven using simple functions. $\endgroup$ – Dosidis Aug 16 '18 at 3:25
  • $\begingroup$ @Dosidis That make sense! Thank you! But I still don't know how to use the inequality.... $\endgroup$ – Lev Ban Aug 16 '18 at 18:11

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