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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?

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    The idea of the title of a question would be that they reveal something about the content. "Proving measurability" has absolutely nothing to do with your question. Please do make an effort and be more specific.2012-03-17

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$\bf Hint:$ If h'(y) is always positive then $h$ is an increasing function. It follows that for any interval $I=(c,d)\subseteq [a,b], \ h((c,d))=(h(c),h(d))$. In this particular case m(h(I))=m((h(c),h(d))=h(d)-h(c)=\int_{(c,d)}h'. For the general case, use approximations to $\lambda$ by open sets.

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    @Patrick You are right $d-c$ should be $h(d)-h(c)$ and the result follows from the Fundamental Theorem of Calculus.2012-03-17