Let \((A,\mu )\) and \((B,\nu )\) be probability spaces, let \(f:A\times B\to [0,\infty ]\) be measurable, and let \(r\ge 1\). Then \((\int _B(\int _A f(x,y)\, d\mu (x))^r\, d\nu (y))^{1/r}\le \int _A(\int _B f(x,y)^r\, d\nu (y))^{1/r}\, d\mu (x)\), with nonnegative extended integrals.

Let \((A,\mu )\) and \((B,\nu )\) be probability spaces, let \(g:A\times B\to [0,\infty ]\) be measurable, and let \(0\lt p\le q\lt \infty \). Then \((\int _B(\int _A g^p\, d\mu )^{q/p}\, d\nu )^{1/q}\le (\int _A(\int _B g^q\, d\nu )^{p/q}\, d\mu )^{1/p}\).

Let \(\mu ,\nu \) be probability measures on \(\mathbb {R}\), \(1\le p\le q\lt \infty \), and \(\rho \in \mathbb {R}\). Suppose \(\| a+\rho bx\| _{L^q(\theta )}\le \| a+bx\| _{L^p(\theta )}\) for every \(a,b\in \mathbb {R}\) and each \(\theta \in \{ \mu ,\nu \} \). Then \(\| a+\rho b(x+y)\| _{L^q(\mu \times \nu )}\le \| a+b(x+y)\| _{L^p(\mu \times \nu )}\) for every \(a,b\).

Let \(f\) be an almost everywhere strongly measurable real function on an arbitrary measure space \((A,\mu )\), let \(C\in [0,\infty ]\) and \(N\in \mathbb N\). If \(\| f\| _n\le C\) for every integer \(n\ge N\), then \(\| f\| _\infty \le C\).

Let \(\mu ,\nu \) be probability measures on \(\mathbb {R}\), \(1\le p\le q\le \infty \), and \(\rho \in \mathbb {R}\). Suppose \(\| a+\rho bx\| _{L^q(\theta )}\le \| a+bx\| _{L^p(\theta )}\) for every \(a,b\in \mathbb {R}\) and each \(\theta \in \{ \mu ,\nu \} \). Then \(\| a+\rho b(x+y)\| _{L^q(\mu \times \nu )}\le \| a+b(x+y)\| _{L^p(\mu \times \nu )}\) for every \(a,b\), with the essential supremum at infinity.