Let $\mu$ be a finite measure on $X$ and $f : X \to \mathbb{R}$ be measurable. Show that if $\int_{X \times X} (f(x) - f(y))^2 d(\mu(x) \times \mu(y)) < \infty $ then $\int_{X} f(x)^2 d\mu(x) < \infty$
If I understand correctly, by Fubini theorem the slice $\int_{X} (f(x) - f(y))^2 d\mu(x) < \infty$ for almost every $y$. Now I tried to get some good estimation on $\int_{X} f(x)^2 d\mu(x)$, but I couldn't do it. Thanks for help
Hint: pick a $y$ for which $\int_{X} (f(x) - f(y))^2 d\mu(x)$ is finite and use the inequality $$ f(x)^2=\left(f(x)-f(y)+f(y)\right)^2\leqslant 2\left(f(x)-f(y)\right)^2+ 2f(y)^2.$$