Two-sample U-statistic parameter

Let $(\mathsf{X},\mathcal{A})$ be a measurable space, let $P,Q$ be probability measures on it, and let $r\ge 1$ be a fixed integer. Give ${\mathsf{X}}^{2r}$ the product sigma-algebra and the product probability measure $P^{\otimes r}\otimes Q^{\otimes r}$. A two-sample order-$(r,r)$ kernel is a measurable function $h:{\mathsf{X}}^{2r}\to \mathbb{R}$ that is symmetric in its first $r$ arguments and separately symmetric in its last $r$ arguments. If $h$ is integrable under this product law, its two-sample U-statistic parameter is

$\theta (P,Q;h):={\int }_{{\mathsf{X}}^{2r}}hd(P^{\otimes r}\otimes Q^{\otimes r})\mathrm{.}$

The value is finite and real. This definition imposes no null hypothesis and no antisymmetry condition; those are hypotheses of later results.