Exercise 9, § 8

Let $u$ and $v$ be two increasing and right-continuous numerical functions on $\mathbf{R}$, such that $u(x) = v(x) = 0$ for $x < 0$. Let $w$ be the increasing right-continuous function on $\mathbf{R}$ defined by $w(t) = u(t)v(t)$ for $t \geq 0$, $w(t) = 0$ for $t < 0$; let $\lambda, \mu, \nu$ be the Stieltjes measures associated with $u, v, w$ respectively (§6, Exer. 5).

To every function $f$ defined on $\mathbf{R}$, with values in $\overline{\mathbf{R}}$ or in a Banach space $F$, one makes correspond the function $\bar{f}$ defined on $\mathbf{R}^2$ by the conditions $\bar{f}(x, y) = f(x)$ if y < x , $\bar{f}(x, y) = f(y)$ if $y \geq x$. Show that, for $f$ to be $\nu$-integrable, it is necessary and sufficient that $\bar{f}$ be integrable for the product measure $\lambda \otimes \mu$, in which case $\int f\, d\nu = \iint \bar{f}\, d\lambda\, d\mu$ (prove it first for the characteristic functions of intervals). From this, deduce the formula

In particular, if $u$ and $v$ are continuous on $\mathbf{R}$, one obtains the formula for integration by parts