= Weighted dilation kernel
{title2=$G(x,y)=a(x)f(b(x)y)$}
Let $f$ be a <Borel measurable function> in <L1 space> on $\mathbb R$, and let $a,b$ be measurable real functions of $x$ with $b(x)>0$ and $W=\int_{\mathbb R}|a(x)|/b(x)\,dx<\infty$. The joint integrand $G(x,y)=a(x)f(b(x)y)$ is measurable, and <Tonelli's theorem> followed by <Lebesgue substitution on an interval> gives
$$
\|G\|_{L^1(\mathbb R^2)}=W\|f\|_{L^1(\mathbb R)}.
$$
For each fixed $x$, substitute $z=b(x)y$ in the inner $y$ integral. The absolute Jacobian is $1/b(x)$, leaving $|a(x)|\|f\|_1/b(x)$. Integrating in $x$ proves the formula and <absolute integrability>. <Fubini's theorem> then permits both orders of signed integration, giving
$$
\iint_{\mathbb R^2}G(x,y)\,dx\,dy=
\left(\int_{\mathbb R}\frac{a(x)}{b(x)}\,dx\right)
\left(\int_{\mathbb R}f(y)\,dy\right).
$$
The one-dimensional sections are integrable for <almost every> value of the other variable; the theorem does not assert integrability of every section.
Back to article page
