Positivity (Metric) Property of the Functional Sliced Wasserstein Distance

Establish whether the functional sliced Wasserstein distance FSW_r on the space of probability measures over L^2(𝒮) satisfies the positivity property of a metric; specifically, show that FSW_r(P, Q) = 0 implies P = Q for all P, Q ∈ 𝒫(L^2(𝒮)). If positivity holds, derive an invertible Radon-like transformation for probability measures on L^2(𝒮) that enables such a proof.

Background

The paper introduces the functional sliced Wasserstein distance (FSW_r) as an extension of sliced Wasserstein techniques to infinite-dimensional functional data spaces, with a focus on L^2(𝒮) and the spherical case for climate model validation. The authors prove that FSW_r is a pseudometric and satisfies identity, symmetry, triangle inequality, and r-convexity.

However, whether FSW_r is a full metric remains unresolved because the positivity property (FSW_r(P, Q) = 0 implies P = Q) has not been established. The authors note that proving positivity would likely require constructing an invertible Radon-like transform for probability measures on L^2(𝒮), analogous to invertibility conditions used in finite-dimensional sliced Wasserstein settings.