Theorem 16.2.3 (Minkowski’s Inequality for Integrals).label Let $(X, \cm, \mu)$ and $(Y, \cn, \nu)$ be $\sigma$-finite measure spaces, $E$ be a Banach space, and $f: X \times Y \to E$ be $(\cm \otimes \cn)$-strongly measurable. Then:
- (1)
If $E = \real$ and $f \ge 0$, then for any $p \in [1, \infty)$,
\[\braks{\int_{X}\paren{\int_Y f(x, y)\nu(dy)}^{p}\mu(dx)}^{1/p}\le \int_{Y}\norm{f(\cdot, y)}_{L^p(X, \mu; E)}\nu(dy)\] - (2)
For any $p \in [1, \infty]$, if
- (a)
For $\nu$-almost every $y \in Y$, $f(\cdot, y) \in L^{p}(\mu; E)$.
- (b)
The mapping $y \mapsto \norm{f(\cdot, y)}_{L^p(\nu; E)}$ is in $L^{1}(\mu; \real)$.
then the mapping $x \mapsto \int_{Y} f(x, y)\nu(dy)$ is in $L^{p}(\mu; E)$, with
\[\norm{\int_Y f(\cdot, y)\nu(dy)}_{L^p(X, \mu; E)}\le \int_{Y}\norm{f(\cdot, y)}_{L^p(X, \mu; E)}\nu(dy)\]
Proof, [Theorem 6.19, Fol99]. (1): If $p = 1$, then by Tonelli’s Theorem,
Now suppose that $p \in (1, \infty)$, assume without loss of generality that $\int_{Y}\norm{f(\cdot, y)}_{L^p(X, \mu; E)}\nu(dy) < \infty$. Let $q$ be the Hölder conjugate of $p$ and $\phi \in L^{q}(\mu; \real) \cap L^{+}(\mu)$ with $\phi \ge 0$, then by Tonelli’s Theorem,
As $f \ge 0$ and the above holds for all $\phi \in L^{q}(\mu; \real) \cap L^{+}(\mu)$, Theorem 16.3.3 implies that $x \mapsto \int_{Y} f(x, y)\nu(dy) \in L^{p}(\mu)$ with
$\square$
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