Minkowski's Inequality for Double Integrals

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

Let $\struct {X \times Y, \Sigma \otimes \Sigma', \mu \times \nu}$ 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 $p \ge 1$:


 * $\displaystyle \paren {\int_X \paren {\int_Y \size {\map f {x, y} } \map {\rd \nu} y}^p \map {\rd \mu} x}^{1/p} \le \int_Y \paren {\int_X \size {\map f {x, y} }^p \map {\rd \mu} x}^{1/p} \map {\rd \nu} y$

Also see

 * Minkowski's Inequality for Integrals