Let $F, G: {\bf R} \rightarrow {\bf R}$ be monotone non-decreasing and continuous. Show that $$\displaystyle \int_{[a,b]} F\ dG = - \int_{[a,b]} G\ dF + F(b) G(b) - F(a) G(a)$$ for any compact interval $[a,b]$.\par This formula can be partially extended to the case when one or both of $F, G$ have discontinuities, but care must be taken when $F$ and $G$ are simultaneously discontinuous at the same location.

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An Introduction to Measure Theory

Exercise 1.7.17 (Integration by parts formula)

Let $F, G : R \to R$ be monotone non-decreasing and continuous. Show that

\int_{[a, b]} F dG = - \int_{[a, b]} G dF + F (b) G (b) - F (a) G (a)

for any compact interval $[a, b]$ .

This formula can be partially extended to the case when one or both of $F, G$ have discontinuities, but care must be taken when $F$ and $G$ are simultaneously discontinuous at the same location.

Exercise 1.7.17 (Integration by parts formula)

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