Actually it is quicker to sketch the proof than checking a reference.
Assume $u:\mathbb{R}^n\to \mathbb{R}$ is measurable and periodic wrto $x_i$ with period $b_i - a_i$, for $1\le i\le n$. Then by a linear change of variables $\|u\| _ {p,M}=\|u _ \nu \| _ {p, M} $ for $1\le p\le\infty$. So if $u\in L^p_{loc}$ the sequence $\{u _ \nu \} _ \nu $ is bounded in $L^p(M)$. As a general fact, when checking the weak (or weak*) convergence of a bounded sequence in a Banach space $E$ (resp., in its dual), a norm-dense set of test element of $E^*$ (resp. $E$) is sufficient. Here the thesis easily follows using as test functions continuous functions on $M$ , for which it holds $$\Bigg | \int_M u _ \nu(x) \phi(x)dx - \int_M \tilde u \phi(x)dx \Bigg | \leq 2\|u\|_{1,M}\\ \omega( {\delta}/{\nu})\\ ,$$
where $\omega$ is a modulus of continuity for $\phi$ on $M$ and $\delta$ is the diameter of $M$. Note that, of course, the argument wouldn't work for $p=1$, as continuous functions are not dense in $L^\infty$.