$\lim_{\lambda \to \infty}\dfrac{1}{\lambda}\int_0^{\lambda}yf(y)dy = 0$?

Question

Assume that $f : [0,\infty) \to \mathbb{R}$ is a Borel--measurable function. Assume also that is integrable with respect to the Borel measure on $[0,\infty)$.

Is it true that: $$\lim_{\lambda\to\infty}\dfrac{1}{\lambda}\int_0^{\lambda}yf(y)dy = 0?$$

I think that it is true, but could not prove it. I apprecaite any hint.

Answer 1

Hint: Let $F(y)=\int\limits_0^{y} f(t)dt$. Then $\frac 1 {\lambda} \int\limits_0^{\lambda} yf(y)dy=\frac 1 {\lambda} [yF(y)|_0^{\lambda} - \int\limits_0^{\lambda} F(y)dy] \to \int_0^{\infty} f(t)dt -\int_0^{\infty} f(t)dt =0$.

[If $g(x) \to c$ as $x \to \infty$ then $\frac 1 {\lambda}\int\limits_0^{\lambda} g(t)dt \to c$].