Let $f\colon\Omega\times\Omega\to\mathbb{R}$ be non-negative. Under which hypotheses does the following inequality hold
$$ \left\{ \int \left[ \int f(x,y)\; \mathrm{d}\mu(y) \right]^p \mathrm{d}\mu(x) \right\}^{1/p} \leq \int \left[ \int f^p(x,y)\; \mathrm{d}\mu(x) \right]^{1/p} \mathrm{d}\mu(y) $$
?
I can easily see that (for example) $f$ should be measurable and $\int f(x,y)\; \mathrm{d}\mu(y)$ should be $L^p$ wrt $x$, what else?
What is required is for $f \ge 0$ to be measurable with respect to the product measure $\mu \times \mu$. All of the integrals make sense even if their values are infinite.