Coincidence of analog and distributional invariants on metrizable spaces

Prove that for every metrizable topological space X and every integer r ≥ 1, the analog sectional category acat(X) and the sequential analog topological complexities ATC_r(X), defined via continuous selections of finite-support probability measures, coincide with the distributional category and distributional topological complexities introduced by Dranishnikov–Jauhari using the Lévy–Prokhorov metric; that is, establish equality of the corresponding analog and distributional invariants on the class of metrizable spaces.

Background

The paper introduces analog versions of sectional category and topological complexity, defined using continuous choices of fiberwise probability measures with finite support. Independently, Dranishnikov and Jauhari proposed closely related "distributional" invariants defined using the Lévy–Prokhorov metric, which a priori requires the underlying space to be metrizable.

The authors note strong similarities between the two frameworks and posit that, at least on metrizable spaces where both definitions apply, the invariants should agree. Establishing this equivalence would unify two approaches and clarify the role of metrizability in these probabilistic formulations of classical homotopy invariants.