Root-L2 Hellinger distance between density laws

Let $(\mathsf{X},\mathcal{A},\mu )$ be a measure space and let $p,q:\mathsf{X}\to [0,\mathrm{\infty })$ be measurable densities with $\int pd\mu =\int qd\mu =1$. They define probability laws $P(B)={\int }_{B}pd\mu$ and $Q(B)={\int }_{B}qd\mu$ (Probability measures and probability spaces). Their root-L2 Hellinger distance, with the root-L2 normalization, is $h(P,Q):={({\int }_{\mathsf{X}}(\sqrt{p}-\sqrt{q}{)}^{2}d\mu )}^{1\mathrm{/}2}\mathrm{.}$

The dominating measure $\mu$ is fixed throughout a use of this definition. Changing either density on a $\mu$-null set does not change the value, since the integrand changes only on a null set. This definition does not assert that separate choices of density versions are jointly measurable in an additional parameter.

The integrand is measurable: square root is Borel on $[0,\mathrm{\infty })$, and subtraction and squaring preserve measurability (Composition with a Borel measurable outer map preserves measurability, Arithmetic and lattice operations preserve measurability whenever they are defined). For $a,b\ge 0$, $(\sqrt{a}-\sqrt{b}{)}^2=a+b-2\sqrt{ab}\le a+b$. Thus $0\le (\sqrt{p}-\sqrt{q}{)}^2\le p+q$ pointwise; monotonicity and additivity of the nonnegative integral (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral) give $h(P,Q{)}^2\le \int (p+q)d\mu =2$, so the defining integral is finite. The set functions $B\mapsto {\int }_{B}pd\mu$ and $B\mapsto {\int }_{B}qd\mu$ are measures by The measure with density $f$ relative to $\mu$, and their total masses are one, so they are probability laws by Probability measures and probability spaces. Changing either density on a $\mu$-null set changes the integrand only on a null set, whose nonnegative integral vanishes (A nonnegative integral over a null set vanishes), confirming the stated version invariance.