I am currently calculating $$\lim_{n \to \infty} \int_{[0,1]^n} f\left(\frac{x_1+...+x_n}{n}\right)\mathrm{d}x_1...\mathrm{d}x_n$$ with $f$ a continuous function between $[0,1]$.
We deduce that $$\int_{[0,1]^n} f\left(\frac{x_1+...+x_n}{n}\right)\mathrm{d}x_1...\mathrm{d}x_n=\mathbb{E}\left[f\left(\frac{X_1+...+X_n}{n}\right)\right]$$ where $X_1,...,X_n$ are independent and uniforme on $[0,1]$ random variable.
With the strong law of large numbers, $\frac{X_1+...+X_n}{n}$ converge toward $\frac{1}{2}$.
However, I don't reach to understand how to use the dominated convergent theorem. Indeed, I don't know how to deal with the fact that as $n$ increase, the number of integral increase too in the expression $ \int_{[0,1]^n} f\left(\frac{x_1+...+x_n}{n}\right)\mathrm{d}x_1...\mathrm{d}x_n$.
Thanks a lot for your help.
If $f:[0,1]\to \mathbb R$ is continuous, it is also bounded. Thus $\int f d\mu_n \to \int f d\mu$ for any sequence of probability measures $\mu_n$ converging weakly to $\mu$. In particular, one has $$\lim_{n\to\infty}\int_{[0,1]^n}f\left(\frac{x_1+\dots+x_n}{n}\right)dx_1\dots dx_n = f(1/2)$$