Suppose that $\displaystyle F$ is continuous and increasing on [a,b], with $\displaystyle F(a)=c, F(b)=d$. Suppose that $\displaystyle f$ is integrable, show that

$\displaystyle \int_c^df(y)dy=\int_a^bf(F(x))\mu_F(x)$ where $\displaystyle \mu_F((x,y])=F(y)-F(x)$

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So far I have shown that $\displaystyle m(E)=\mu_F(F^{-1}(E))$ for Borel sets $\displaystyle E$, but I don't see how this helps.