The problem I'm having trouble answering is: Suppose $h$ is a differentiable function on $[a,b]$ with a continuous, positive derivative h'(y) for all $x \in [a,b]$. For a measurable subset $\lambda\subset[a,b]$, show that m(h(\lambda)) = \int_{\lambda}h'. Then, use this to prove the Integration by substitution formula, namely that \int_a^bf(g(x))g'(x)dx = \int_{g(a)}^{g(b)}f(t)dt.
Does anyone have any suggestions on how I should go about solving this?