Exercise 11, § 8

Integration, V, p. 117

Let $p$ be a finite real number $\geq 1$ , T and X two locally compact spaces, $\lambda$ a positive measure on T, $\mu$ a positive measure on X, and $f$ a function $\geq 0$ defined on $T \times X$ , integrable along with $f^p$ for the measure $\lambda \otimes \mu$ . Prove the inequality

\left( \int^* \left( \int f\, d\mu \right)^p d\lambda \right)^{1/p} \leq \int^* \left( \int f^p\, d\lambda \right)^{1/p} d\mu.

(For every $t \in T$ , apply Hölder’s inequality to the function $x \mapsto f(t, x)$ , put into the form

f(t, x) = g(t, x) \left( \int f^p(t, x)\, d\lambda(t) \right)^{1/pq},

where $q$ is the exponent conjugate to $p$ .) Show that equality cannot hold unless $f$ is equivalent to a function of the form $g \otimes h$ .

Solution

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