Let $U:=\prod_{i}^{d}(a_{i},b_{i})$ a subset of $\mathbb{R}^d$ with $a_{i}<b_{i}$ for each i and let $f\in{L^{p}(U)}$ for some $1<p<\infty$. Now extend $f$ periodically to the whole $\mathbb{R}^d$ (with $U$ as a basic period) and set $f_{k}(x):=f(kx)$, for $k \in \{1,2,\ldots\}$. I have to prove that $$ f_{k}\rightharpoonup\hat{f}=\frac{1}{|U|}\int_{U} f(x)dx \quad \text{in } L^{p}(U). $$
Actually, I don't know how to proceed. Maybe I don't have understood well the point about the periodic extension of $f$. Is that someone that could give me a hint? Thank you very much for your help!
This is a so called Averaging Lemma. If you look in a basic homogenization textbook you can find this proven.
Heuristically, your space $U\approx \cup\Pi^k$ for large $k$, where $\Pi^k$ captures each period of $f_k$ (so that larger $k$ have smaller periods and thus better approximate $U$). On each of these $\Pi^k$, nice functions are essentially constant and so you should consider working with Riemann sums to get appropriate estimates.
To help yourself, prove the result on $C^\infty_0(U)\subset L^p(U)$, a dense subset, and convince yourself that this is sufficient to prove weak convergence on all of $L^p$.