I'm strugging to show the change of variable formula in Lebesgue integral. My original question is in here.
I've found a similiar question in here, and the author says $G$ is differentiable a.e, which means $f(g(t))g'(t)$ is integrable. The thing is, I have a difficulty showing this statement. Every other statement is understandable, except for the fact that $f(g(t))g'(t)$ is integrable. The whole explanation make sense under the integrability of $f(g(t))g'(t)$ I guess. But I don't think letting $g(t) = y$ and differentiating $y$ with $t$ is proper way, (my first idea).
Any other comments about the integrability of $f(g(t))g'(t)$ would be grateful. Thank you.