Exercise 5, § 3

Let $f\in L^{\infty}([0,+\infty ([)$ and $k\in \mathbf{R}^*_+$. If

lim $ef(t)dt=\ell$

$x\rightarrow +\infty x$ 0

then

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

(Use the mappings $g(t) =e^{-t}$, and $h(t) =k(1-t)^{k-1}$ for $0\leqslant t\leqslant 1$, $h(t) = 0$ for $t >1$.)