Let $U\subset\mathbb R^n$ be an open set, $T(x)=x+v\,$ is a translation in $\mathbb R^n$ and $f:\,T(U)\longrightarrow\mathbb R$ is Riemannian integrable. I want to show that $\displaystyle\intop_Uf$ is invariant through $T$. In other words, I am trying to prove the change of variable Theorem for the case where the diffeomorphism is a translation \begin{align} \intop_{T(U)}f&=\intop_U(f\circ T)\big|\det(J_T)\big| \end{align} which is written in this case as \begin{align} \intop_{U+v}f = \intop_Uf\circ T. \end{align} My attempt :
Let $I\subset\mathbb R^n$ is a $n-$rectangle contains $U$ and $U+v$. Then by definition, \begin{align} \intop_{U+v}f&=\intop_I\textbf 1_{U+v}\cdot f \\ \intop_Uf\circ T&=\intop_I\textbf 1_{U}\cdot (f\circ T) \end{align} But first, I need to show the integrability of $\textbf 1_{U}\cdot (f\circ T)$. Since $f$ is integrable over $U+v$, it is continuous a.e on $U+v$. We also have the continuousness of $T$ on $U$, thus $f\circ T$ is continuous a.e on $U$, which implies that it is also integrable over $U$ (I'm not sure if my deduce is valid).
Next, I guess my proof will be using the upper and lower sum of $ \textbf 1_{U}\cdot (f\circ T)$. I know that \begin{align} U(\textbf 1_U,P)&=\sum_{\substack{R\in H(P) \\ R\cap U\ne\varnothing}}|R| \\ L(\textbf 1_U,P)&=\sum_{\substack{R\in H(P) \\ R\subset U}}|R| \end{align} but I have no clue how to expand the upper-lower sum to $ \textbf 1_{U}\cdot (f\circ T)$ because this function does not appear necessary to be non-negative.
So how should i keep on ? Could anyone give me some help ? Thanks.