Exercise 8, § 3

Let $g$ be a function satisfying the hypotheses of Exercise 3. Moreover, suppose $g\geqslant 0$ and lim inf$_{x\rightarrow 0}g(x)>0$. Let $f$ be a real measurable function bounded below on $]0,+\infty [$. If

lim$_{x\rightarrow+\infty}\frac{1}{x}\int_0^{+\infty}g(\frac{t}{x})f(t)dt=\ell$

then

tends to $\ell$ when $x$ tends to $+\infty$. (Prove that $\sigma$ is bounded for $x\geqslant 1$, then that $\sigma$ is slowly decreasing, then that

tends to $\ell$ when $x$ tends to $+\infty .)$