Minkowski's Inequality for Double Integrals

Theorem
Let $\left({X, \Sigma, \mu}\right)$ and $\left({Y, \Sigma', \nu}\right)$ be $\sigma$-finite measure spaces.

Let $\left({X \times Y, \Sigma \otimes \Sigma', \mu \times \nu}\right)$ be their product measure space.

Let $f: X \times Y \to \overline \R$ be a $\Sigma \otimes \Sigma'$-measurable function.

Then, for all $p \in \R$ with $p \ge 1$:


 * $\displaystyle \left({\int_X \left({\int_Y \left\vert{f \left({x, y}\right)}\right\vert \, \mathrm d \nu \left({y}\right)}\right)^p \, \mathrm d \mu \left({x}\right)}\right)^{1/p} \le \int_Y \left({\int_X \left\vert{f \left({x, y}\right)}\right\vert^p \, \mathrm d \mu \left({x}\right)}\right)^{1/p} \, \mathrm d \nu \left({y}\right)$

Also see

 * Minkowski's Inequality for Integrals