Let be $u:\Omega \rightarrow \mathbb{R}^n $ a numerical measurable function. Let be $\tau_y u(x) = u(x-y)$. Then, the support($\tau_y u$)=$y$+support($u$).
I am trying: If $x-y = c$ then $\tau_y u(x) = u(x-y)\iff \tau_y u(c+y) = u(c)$. Then support($\tau_y u$) $=\{c+y \in \Omega; u(c)\neq 0\}$. I think that $\{c+y \in \Omega; u(c)\neq 0\} = y + \{c \in \Omega; u(c)\neq 0\}$, but I don't know making a proof?