When we define the convolution on $L^1(\mathbb{R}^n)$, we are interested to proof that $\forall f,g \in L^1(\mathbb{R}^n)$ then $f\star g \in L^1(\mathbb{R}^n)$. In the proof of this we want use the Fubini-Tonelli for said that:
$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|u(x-y)v(y)|dydx = \\ =\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|u(x-y)v(y)|dxdy$
But for that is sufficient to say that:
$u(x-y):(\mathbb{R}^n\times \mathbb{R}^n, \sigma(\mathscr{L}^n\times\mathscr{L}^n)) \rightarrow (R,\mathscr{B}(\mathbb{R}))$
is measurable. However i don't how ti proof this fact.