Root-L2 Hellinger distance for dominated laws

Let $(\mathsf{X},\mathcal{A})$ be a measurable space and let $P,Q$ be probability measures (Probability measures and probability spaces) on it, both dominated by a specified common sigma-finite positive measure $\mu$ as in Dominated statistical experiments. For supplied finite-valued nonnegative measurable density representatives $p=dP\mathrm{/}d\mu$ and $q=dQ\mathrm{/}d\mu$ in the convention of Likelihood densities, the root-L2 Hellinger distance is

$h(P,Q):={(\int (\sqrt{p}-\sqrt{q}{)}^{2}d\mu )}^{1\mathrm{/}2}\mathrm{.}$

The square roots are measurable: for every real $r$, $\{\sqrt{p}>r\}$ is the whole space if $r<0$ and equals $\{p>r^2\}$ if $r\ge 0$, and the same holds for $q$ (Threshold characterisations of real-valued and extended-real-valued measurability). The arithmetic and lattice rules make the nonnegative integrand measurable (Arithmetic and lattice operations preserve measurability whenever they are defined). It is integrable because $(\sqrt{p}-\sqrt{q}{)}^2\le p+q$ and the integral is additive on nonnegative functions (Additivity of the nonnegative Lebesgue integral), so its integral is at most $P(\mathsf{X})+Q(\mathsf{X})=2$.

The generic density-selection convention Likelihood densities assumes the Axiom of Choice The Axiom of Choice. When $p$ and $q$ are explicitly given, this displayed integral requires no further choice.