= L1 norm identity for a translated kernel product
{c}
{title2=$\|k(x-y)g(y)\|_{L^1(\mathbb R^2)}=\|k\|_1\|g\|_1$}
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> give
$$
\int_{\mathbb R^2}|k(x-y)g(y)|\,dx\,dy
=\int_{\mathbb R}|g(y)|\left(\int_{\mathbb R}|k(x-y)|\,dx\right)dy
=\|k\|_1\|g\|_1.
$$
Finiteness 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 $\|k*g\|_1\leq\|k\|_1\|g\|_1$.
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