I'm trying to show the following:
If $f \in L^1(\mathbb{R})$ with $f$ nonnegative, then $$\lim_{n \to \infty}\frac{1}{n}\int_{-n}^{n}tf(t)dt=0$$
I"ve shown that for every $n≥0$ we have $$ \frac{1}{n}\int_{-n}^{n}tf(t)dt≤\int_{-n}^{n}f(t)dt$$ but I'm not sure if that's useful or not. My aim is to employ one of the standard convergence theorems, but I'm not sure how to set it up so far.
Note that $f_{n}(t)=\frac{1}{n} tf(t)\mathbb{1}_{[-n,n]}(t)$ goes to $0$ pointwise and that for every $t$, we have $|f_{n}(t)| \le |f(t)| \in L^{1}(\mathbb{R})$ because $f_{n}(t)=0$ for $|t|>n$. Then use dominated convergence.