Exercise 11, § 6

The notations are those of No. 5; $g$ is a locally $\mu$-integrable numerical function on $I$.

a) For $G$ to be a $\lambda$-proper mapping of $I$ into $G(I)$, it is necessary and sufficient that the following conditions be fulfilled: $1^\circ$ the limits $G(a+)$ and $G(b-)$ exist in $\overline{\mathbf{R}}$; $2^\circ$ if $G(a+) \in G(I)$ then $g$ is $\mu$-integrable on the interval $]a, x_0[$; $3^\circ$ if $G(b-) \in G(I)$ then $g$ is $\mu$-integrable on the interval $]x_0, b[$.

b) Suppose that the limits $G(a+)$ and $G(b-)$ exist in $\overline{\mathbf{R}}$. Show that if $f$ is such that $t \mapsto f(G(t))g(t)$ is integrable for Lebesgue measure on $I$, then $f$ is integrable on the interval with end-points $G(a+)$ and $G(b-)$ for Lebesgue measure, and formula (13) holds. (Reduce to the case that $x_0$ is one of the numbers $a, b$; if, for example, $x_0 = a$, observe that there exists in $I$ a strictly increasing sequence $(b_n)$ tending to $b$, such that the sequence $(G(b_n))$ is either increasing or decreasing; apply Prop. 4 of Ch. IV, §4, No. 3 and Lebesgue’s theorem).