I have the following problem that I am trying to work through.
For a non-negative integrable function $f$ over an interval $[c,d]$ and a strictly increasing absolutely continuous function $g$ on $[a,b]$ such that $g([a,b])\subset [c,d]$, is it possible to justify the change of variables formula $$\int_{g(a)}^{g(b)}f(y)\:dy=\int_{a}^{b}f(g(x))g'(x)\:dx$$ by showing that for almost all $x\in(a,b)$, we have $$\frac{d}{dx}\left[\int_{g(a)}^{g(x)}f(s)\:ds-\int_{a}^{x}f(g(t))g'(t)\:dt\right]=0?$$
I think that if you show that differentiation fact, that would possibly justify the change of variables formula (maybe through some facts about absolute continuity?). However, I am completely lost and really have no idea where to begin. Thanks in advance for any help or suggestions!
