On a metric on the space of idempotent probability measures

In this paper we construct a metric on the space of idempotent probability measures on the given compactum, which is an idempotent analog of the Kantorovich metric on the space of probability measures.

💡 Research Summary

The paper addresses the problem of endowing the space of idempotent probability measures on a compact Hausdorff space X with a natural metric that plays the same role as the classical Kantorovich (or Wasserstein‑1) distance does for ordinary probability measures. An idempotent probability measure is defined with respect to the max‑plus algebra: addition is replaced by the maximum operation, scalar multiplication by ordinary addition, and integration becomes a supremum of a function plus the measure of a point. This non‑linear framework appears in tropical geometry, optimal control, and large‑deviation limits, yet a systematic metric structure has been missing.

The authors begin by recalling the necessary background on max‑plus algebra, the definition of idempotent measures, and the weak* topology on the space M_I(X) of such measures. They then introduce the central object of the study, the “idempotent Kantorovich metric” d_I, defined by

 d_I(μ, ν) = sup_{φ∈Lip_1(X)} | ∫_X φ dμ ⊕ (−∫_X φ dν) |,

where Lip_1(X) denotes the set of real‑valued Lipschitz functions on X with Lipschitz constant ≤ 1, ⊕ is the max operation, and the minus sign denotes the additive inverse in the max‑plus sense (i.e., ordinary negation). The absolute value is interpreted as the max‑plus distance between two scalars.

A substantial part of the paper is devoted to proving that d_I satisfies all metric axioms. Symmetry follows from the commutativity of max, non‑negativity and the identity of indiscernibles are shown by constructing a Lipschitz test function that separates distinct measures, and the triangle inequality is derived from the max‑plus subadditivity of Lipschitz functions.

Next, the authors compare the topology induced by d_I with the usual weak* topology on M_I(X). They prove that a sequence {μ_n} converges to μ in d_I if and only if for every continuous φ (in fact, for every Lipschitz φ) the idempotent integrals ∫ φ dμ_n converge to ∫ φ dμ. Consequently, d_I generates exactly the weak* topology, allowing one to transfer all known compactness and continuity results from the classical setting.

The paper then establishes completeness: any Cauchy sequence with respect to d_I has a limit in M_I(X). The argument proceeds by showing that for each Lipschitz φ the scalar sequence of integrals is Cauchy in ℝ, defining a limit functional, and then invoking the representation theorem for idempotent measures to obtain a genuine limit measure. Because X is compact, the authors also prove that (M_I(X), d_I) is itself compact, essentially by an Ascoli‑type argument adapted to the max‑plus context.

To illustrate the construction, the authors treat two concrete families of examples. For a finite discrete space X = { x_1,…,x_k }, an idempotent probability measure is simply a vector (a_1,…,a_k) with max a_i = 0, and d_I reduces to the supremum norm distance sup_i |a_i − b_i|. For the unit interval X =