I'm struggling to compute the (integral) mean value of a piecewise affine function in $H^1_0(\Omega)$.
The setting is the following: let $f$ be a function defined as $f= \sum_{j=1}^{N}(\mu^j \cdot x) \chi_{S_j}$, where $S_j$ is a partition of $\mathbb{R}^n$ into simplexes such that $S_j \cap \Omega \neq \emptyset$. Therefore $\nabla f= \mu^j$, over each $S^j$, with $j=1,...,N$.
It is well known that the mean value of a (generic) function $g:\Omega \rightarrow \mathbb{R}^n$, with $\Omega \subset \mathbb{R}^n$, is defined as
$$\bar{g}=\frac{1}{|\Omega|} \int_{\Omega} g(x)dx$$
Denote by $C^k_\epsilon=\epsilon(C+v^k)$, where $C=[0,1]^n$ is the $n$-dimensional periodic cube covering $\mathbb{R}^n$ and $v^k$ is an $n$-dimensional integer. Then, if we want to compute the mean value of $f$ over $C^k_\epsilon$ we have to calculate the integral
$$\frac{1}{|C^k_\epsilon|} \int_{C^k_\epsilon} f(x)dx= \frac{1}{\epsilon^n} \int_{C^k_\epsilon} f(x)dx=$$
at this stage I want to write the last integral over $C$, then I have to change the variables $x/\epsilon=y$, thus $dx=\epsilon^n dy$, then
$$=\frac{1}{\epsilon^n} \int_{C^k_\epsilon} f(x)dx= \frac{1}{\epsilon^n} \epsilon^n \int_{C} f(\epsilon y)dy=\int_{C} f(\epsilon y)dy$$
at this point I get stuck because the result must be $f(\epsilon v^k)$ but I can't write it formally correct. Somebody can help me?