Let $E$ be a $\mathbb R$-vector space, $\mathcal E$ be a translation-invariant $\sigma$-algebra on $E$ (i.e. $B+x\in\mathcal E$ for all $x\in\mathcal E$) and $X,Y$ be $(E,\mathcal E)$-valued independent random variables on a probability space $(\Omega,\mathcal A,\operatorname P)$.
Are we able to show that $$\operatorname P\left[X\in A,X+Y\in B\right]=\int_A\operatorname P\left[X\in{\rm d}x\right]\operatorname P\left[Y\in B-x\right]\tag1$$ for all $A,B\in\mathcal E$?
The result seems so obvious that I don't find a way to prove it.