L1 norm identity for a translated kernel product ($\Vert k(x-y)g(y){\Vert }_{L^1({\mathbb{R}}^2)}=\Vert k{\Vert }_1\Vert g{\Vert }_1$)

Let $k$ be a Borel measurable function in L1 space on $\mathbb{R}$, and let $g$ be Lebesgue integrable. The function $f(x,y)=k(x-y)g(y)$ is measurable. Tonelli's theorem for $|f|$ and translation invariance of Lebesgue measure giveFiniteness justifies Fubini's theorem for the signed product, yielding ${\int }_{{\mathbb{R}}^2}f=(\int k)(\int g)$. In particular an integrable odd function $k$ makes this signed total zero while the joint L1 norm can remain positive. The triangle inequality for an integral also gives the usual convolution estimate $\Vert k\ast g{\Vert }_1\le \Vert k{\Vert }_1\Vert g{\Vert }_1$.