Showing $\lim_{n \to \infty}\int_0^1x^nf(x)\,d \mu =0$ Let $f:[0,1] \to [0, \infty]$ be a measurable function s.t. $\int_0^1 f \,d \mu < \infty$. I wanna show that $$\lim_{n \to \infty}\int_0^1x^nf(x)\,d \mu =0$$
I thought to use the dominated convergence theorem but I don't see how to start.
EDIT:
For the next step I should calculate $\int_0^1 \lim_{n \to \infty}x^nf(x) \,d \mu$. This should be zero but how, can I see this the best way? I was thinking about to show that $\lim_{n \to \infty}x^nf(x)=0$ for $x \in [0,1)$, but what is for $x=1$?
 A: If you don't really want to use Lebesgue Dominated Convergence Theorem, you can simply use Monotone Convergence Theorem.
First of all, let us see the following basic fact: For $f_{N}\geq 0$ are such that $f_{N}(x)\downarrow 0$ for every $x$, and $\int f_{1}<\infty$, then $\int f_{N}\rightarrow 0$. This is a matter of Monotone Convergence Theorem if we look at the increasing sequence $(f_{1}-f_{N})_{N=1}^{\infty}$ and obtain that $\int f_{1}-f_{N}\rightarrow\int f_{1}$.
Now let $N$ be fixed. Then
\begin{align*}
\int_{0}^{1}x^{n}f(x)dx&=\int_{0}^{1-1/N}x^{n}f(x)dx+\int_{1-1/N}^{1}x^{n}f(x)dx\\
&\leq\left(1-\dfrac{1}{N}\right)^{n}\int_{0}^{1}f(x)dx+\int_{0}^{1}1_{[1-1/N,1]}(x)f(x)dx.
\end{align*}
Let $f_{N}(x)=1_{[1-1/N,1]}(x)f(x)$, then $f_{N}(x)\downarrow 0$, apply the basic fact just before, for $\epsilon>0$, let $N$ be such that
\begin{align*}
\int_{0}^{1}1_{[1-1/N,1]}(x)f(x)dx<\epsilon.
\end{align*}
Now $1-1/N<1$ and hence for large enough $n$, we have
\begin{align*}
\left(1-\dfrac{1}{N}\right)^{n}\int_{0}^{1}f(x)dx<\epsilon,
\end{align*}
for all such $n$,
\begin{align*}
\int_{0}^{1}x^{n}f(x)dx<2\epsilon.
\end{align*}
A: Hint:
You have the exact right idea with the dominated convergence theorem!
On $[0,1]$ we see $x^n f \to 0$ pointwise almost everywhere
(do you see why? What about if $f = \infty$ at certain points?).
Then, as mentioned in the comments, $|x^n f| \leq |f|$. Can you use this, plus the dominated convergence theorem, to prove that $\int x^n f \to \int 0 = 0$?

I hope this helps ^_^
