$f: \mathbb R^n\to \mathbb R^m$ Lebesgue measurable, is it true that $(x,y)\mapsto f(x+y)$ is Lebesgue measurable on $\mathbb R^{2n}$?
I know in general $f \circ g$ is not Lebesgue measuable provided $f$ Lebesgue measurable and $g$ continuous, but I don't know whether it is true when $g$ is addition.
I'm interested in this question because in the book Measure Theory and Fine Properties of Functions, in Step 7 of the proof of the coarea formula, it is assumed that $(y,w) \mapsto \mathcal H^{n-m}(A\cap f^{-1}(y-\epsilon w))$ is $\mathcal L^{2m}$- measurable, provided $A$$\mathcal L^n$- measurable and $f:\mathbb R^n\to \mathbb R^m $ Lipschitz. However, in the previous section of the book, it is only proved that $y \mapsto \mathcal H^{n-m}(A\cap f^{-1}(y))$ is $\mathcal L^m$- measurable.