Suppose \( (X_1, \mu_1) \) and \( (X_2, \mu_2) \) are two \(\sigma\)-finite measure spaces, and \( F : X_1 \times X_2 \to \mathbb{R} \) is measurable. Then Minkowski’s integral inequality is
\[ \left( \int_{X_2} \left\vert \int_{X_1} F(x,y) \, d\mu_1(x) \right\vert^p d\mu_2(y) \right)^{1/p} \le \int_{X_1} \left( \int_{X_2} |F(x,y)|^p \, d\mu_2(y) \right)^{1/p} d\mu_1(x), \]for \( p \in [1, \infty) \), with obvious modifications in the case \( p = \infty \).
If \( p > 1 \) and both sides are finite, then equality holds if and only if
\[ |F(x,y)| = \varphi(x) \psi(y) \quad \text{a.e.}, \]for some non-negative measurable functions \( \varphi \) and \( \psi \).