Let $f$ be a continuously differentiable function defined on $\mathbb{R}$, $\mu$ a complex (Radon) measure on $\mathbb{R}$, and $F(x)=\mu((-\infty,x])$. Why is it true that $$\int f d\mu=\int f'(x)F(x)dx$$ Folland suggests that one should use the integration by part formula (Theorem 3.36 on his Real Analysis), but I did not see how it works.
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Does this answer your question?(https://math.stackexchange.com/a/3724004/121671) – Mittens Aug 05 '23 at 20:38
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Could you say the theorem /exercise number where that statement is found in Folland's book? – Fubini Aug 15 '23 at 12:59