Lower semi-continuity of the Hellinger-Fisher-Rao distance

Question

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance $$ H^2(\rho,\mu)=\int_{\Omega}\left|\sqrt{\frac{d\rho}{d\lambda}}-\sqrt{\frac{d\mu}{d\lambda}}\right|^2 d\lambda $$ shows up. Here $\Omega\subset R^d$ is a (possibly unbounded) smooth domain, $\rho,\mu$ are nonnegative Borel measures with finite mass on $\Omega$, $\lambda$ is any reference (nonnegative, Borel) measure on $\Omega$ such that $\rho,\mu$ are simultaneously absolutely continuous with respect to $\lambda$, and $\frac{d\rho}{d\lambda},\frac{d\mu}{d\lambda}$ denotes the corresponding Radon-Nikodym derivatives. Note that by 1-homogeneity this definition does not depend on the choice of the reference measure $\lambda$. This distance is ususally defined for probability measures (i-e with fixed unit mass $|\rho|=|\mu|=1$) but still gives a distance on the set $\mathcal M^+(\Omega)$ of nonnegative Borel measures with finite mass (thus possibly $|\rho|\neq |\mu|$).

Question: is it true that $H$ is lower semi-continuous with respect to the weak-$\ast$ convergence of measures, $$ \rho^n\overset{\ast}{\rightharpoonup}\rho,\mu^n\overset{\ast}{\rightharpoonup}\mu \quad\Rightarrow\quad H(\rho,\mu)\leq \liminf \,H(\rho^n,\mu^n)??? $$

Let me remind that the weak-$\ast$ convergence of measures is defined by duality with continuous compactly supported functions $\phi\in \mathcal C^\infty_c(\Omega)$. Also, I would be happy if this were true for the (stronger) weak $L^1(\Omega,dx)$ convergence for absoluetly continuous measures (with respect to, say, Lebesgue's measure $dx$) instead of weak-$\ast$ convergence of measures.

Answer 1

The lower semicontinuity of the Hellinger-Rao distance follows from the following more general lower semicontinuity result (see e.g. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations, Thm. 3.4.1):

If $f\colon \mathbb{R}^n\to [0,\infty]$ is a proper lower semicontinuous convex positively $1$-homogeneous function, then the functional $$ F\colon \mathcal{M}(\Omega)^n\to [0,\infty],\,\mu\mapsto \int_\Omega f\left(\frac{d\mu^1}{d\lambda},\dots,\frac{d\mu^n}{d\lambda}\right)\,d\lambda $$ is sequentially weak* lower semicontinuous. As in the question, $\lambda$ is any scalar measure on $\Omega$ such that $\mu^1,\dots,\mu^n\ll\lambda$.

Let me sketch the proof. First note that $f$ is the supremum of a sequence of affine-linear functions $f_i$. By monotone convergence, $$ \int_\Omega \sup_{1\leq i\leq N}f_i^+\left(\frac{d\mu}{d\lambda}\right)\,d\lambda\to F(\mu),\;N\to \infty. $$ For each $N\in\mathbb{N}$ we can decompose $\Omega$ into disjoint subsets $E_j$ such that $f_j^+=\sup_{1\leq i\leq N}f_i^+$. Thus $$ F(\mu)=\sup\left\lbrace\sum_{i\in I}\int_{E_i}f_i^+\left(\frac{d\mu}{d\lambda}\right)\,d\lambda\,\Bigg|\, I\subset \mathbb{N}\text{ finite}, E_i\cap E_j=\emptyset\text{ for }i\neq j\right\rbrace. $$ Using the regularity of the measures, one can restrict to disjoint open sets $E_i$ in the supremum above. For $U\subset \Omega$ open one has $$ \int_U f_i^+\left(\frac{d\mu}{d\lambda}\right)\,d\lambda=\sup\left\lbrace \int_\Omega f_i\left(\frac{d\mu}{d\lambda}\right)\phi\,d\lambda\,\Bigg|\,\phi\in C_c^\infty(U),\,0\leq \phi\leq 1\right\rbrace. $$ Since $f_i$ is affine-linear, the functional in the supremum on the right side is sequentially weak* continuous. Therefore $F$ is sequentially weak* lower semicontinuous as supremum of sequentially weak* continuous maps.

Remark: I treated the reference measure as fixed. This is justified since for a sequence $(\mu_k)$ one can always take something like $$ \lambda=\sum_{k=1}^\infty\sum_{l=1}^n |\mu_k^l|. $$ For this reason the argument only works for sequential semicontinuity.