Exercise 10, § 8

Let T and X be two locally compact spaces, $\lambda$ a positive measure on T, $\mu$ a bounded positive measure on X such that $\mu(X) = 1$. Let $f$ be a numerical function $> 0$ at every point of $T \times X$, integrable along with $\log f$ for the measure $\lambda \otimes \mu$. Prove the inequality

and show that equality cannot hold unless $f$ is equivalent to a function of the form $g \otimes h$ (apply the inequality of the geometric mean (Ch. IV, §6, Exer. 7 d)), for every $t \in T$, to the function $x \mapsto f(t, x)/(\int f(t, x)\, d\lambda(t))$.