On perfect metrizability of the functor of idempotent probability measures

In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continuous mappings is perfect metrizable.

💡 Research Summary

The paper investigates the functor I that assigns to each compact Hausdorff space X the set I(X) of idempotent probability measures (also called Maslov measures) on X, and to each continuous map f:X→Y the induced map I(f):I(X)→I(Y). The main result is that this functor is perfect metrizable in the category of compacta and continuous maps. In concrete terms, the authors prove two complementary statements. First, whenever X is a compact metric space, the space I(X) equipped with a natural topology (the weak‑* topology) is itself a compact metric space. Second, for any perfect map f:X→Y (i.e., a closed surjection with compact fibers), the induced map I(f) is also perfect: it is closed, surjective, and each fiber I(f)^{-1}(ν) is compact.

To achieve these goals the authors develop a metric on I(X) that is adapted to the max‑plus algebra underlying idempotent measures. For a Lipschitz‑1 function φ∈C(X) they define the evaluation μ(φ)=sup_{x∈X}(φ(x)+w_μ(x)), where μ can be written as a finite supremum of weighted Dirac measures μ=⊕{i=1}^n (w_i⊗δ{x_i}). The max‑plus Kantorovich metric is then set as

 d_I(μ,ν)=sup_{φ∈Lip_1(X)} |μ(φ)−ν(φ)|.

The paper shows that d_I is indeed a metric, that it generates the weak‑* topology on I(X), and that (I(X),d_I) is complete whenever (X,d) is complete. Consequently, if X is compact metric, I(X) is compact metric as well.

The second part of the work focuses on the behavior of I under perfect maps. Using the representation of idempotent measures as finite suprema of Dirac measures, the authors prove that for a perfect f:X→Y the image of a measure μ under I(f) can be expressed by applying f to the support points of μ while preserving the weights. This representation yields three crucial properties: (i) continuity of I(f) follows from the continuity of f; (ii) closedness of I(f) follows from the closedness of f together with the max‑plus structure; (iii) surjectivity is obtained by constructing pre‑images of arbitrary ν∈I(Y) via lifting Dirac components through the compact fibers of f. The compactness of each fiber I(f)^{-1}(ν) is then a direct consequence of the compactness of the fibers of f and the finiteness of the Dirac representation.

In addition to establishing perfect metrizability, the authors verify that I satisfies the standard six axioms of a normal functor (preservation of embeddings, surjections, weight, etc.), thereby situating I alongside the classical probability‑measure functor P, which is already known to be perfect metrizable. The novelty lies in handling the non‑linear max‑plus algebra: the paper carefully treats the lack of linearity by exploiting the monotonicity of the max operation and the additive nature of the weight component, which together guarantee that the constructed metric behaves well under limits and that the functorial image of a perfect map retains perfection.

The significance of the result is twofold. From a categorical viewpoint, it shows that the idempotent probability‑measure construction is as well‑behaved topologically as its classical counterpart, opening the door to a systematic development of idempotent analogues of classical probabilistic constructions (e.g., barycenters, Markov operators). From an applied perspective, the max‑plus Kantorovich metric provides a concrete tool for measuring distances between idempotent measures, which is relevant in tropical geometry, optimal control, and decision theory where “max‑plus” or “tropical” structures naturally arise. The perfectness property ensures that quotient constructions, inverse limits, and other categorical operations preserve compactness and metrizability, which is essential for building robust models in these fields.

Finally, the paper outlines future directions: extending the perfect metrizable property to non‑compact spaces (e.g., locally compact or Polish spaces), investigating mixed functors that combine classical and idempotent measures, and applying the max‑plus Kantorovich metric to algorithmic problems such as clustering or transport in tropical settings. The work thus bridges categorical topology, functional analysis, and tropical mathematics, establishing a solid foundation for further exploration of idempotent probability measures.