Localized Hilbert lift of the feature energy

Assume AC. Fix an arbitrary probability label space $(S,\mathrm{\Sigma },\sigma )$ and $r>0$. Let $K=L^2(\sigma ;{\mathbb{R}}^d)$ with its usual real Hilbert inner product, and let $C_r=\{X\in K:X(s)\in Q_r\text{ for }\sigma \text{-almost every }s\}\mathrm{.}$ Define the indicator ${\iota }_{C_r}$ to be $0$ on $C_r$ and $+\mathrm{\infty }$ outside it. For $X\in C_r$, the compositions $\mathrm{\Phi }\circ X$ and $V\circ X$ are measurable and bounded, and $\mathrm{\Phi }\circ X$ is strongly measurable; hence the scalar integral is finite and the Bochner integral exists. Define $E_r(X)=\{\begin{array}{ll}R\text{\hspace{0.17em}⁣}({\int }_S\mathrm{\Phi }(X(s))d\sigma (s))+{\int }_{S}V(X(s))d\sigma (s),& X\in C_r,\\ +\mathrm{\infty },& X\mathrm{\notin }{C}_r\mathrm{.}\end{array}$ The formula evaluates neither integral when $X\mathrm{\notin }{C}_r$. The space $K$ is Hilbert for every probability label space and is separable when $\mathrm{\Sigma }$ is countably generated. The latter condition holds for finite labels and for Euclidean Borel labels of any dimension, including joint and triple transport-plan labels. For the flow from a law ${\mu }_0\in {\mathcal{P}}_2(\mathrm{\Omega })$ concentrated on $Q_r$, the particular choice is $(S,\mathrm{\Sigma },\sigma )=(\mathrm{\Omega },\mathcal{B}(\mathrm{\Omega }),{\mu }_0)$ and $X_0(s)=s\in C_r\subseteq K$. Values of representatives on $\sigma$-null sets do not affect $C_r$ or $E_r$.

Feature-measure energy and its regularity assumptions

A Euclidean probability label space gives a separable Hilbert L2 space

Bochner-integrable function

Chizat and Bach, Appendix B.2.1

Ambrosio, Gigli, and Savaré, Chapter 4