Minkowski integral inequality
= Minkowski integral inequality
{c}
This is the integral version of the <Minkowski inequality>.
For suitable measurable $F(x,y)$ and $1\leq p<\infty$,
$$
\left\|\int F(x,\mathord\cdot)\,dx\right\|_p
\leq\int\|F(x,\mathord\cdot)\|_p\,dx.
$$
Pair the left side with a unit vector in the dual $L^q$ space, use Tonelli's theorem, and apply Hölder's inequality in $y$.