Spaces of idempotent measures: a survey of results
by
Mykhailo Zarichnyi
Ivan Franko National University of Lviv

The idempotent mathematics is a part of mathematics dealing with some idempotent operations (e.g., ⊕ = max) instead of usual addition and multiplication. In the recent decade, the results and methods of the idempotent mathematics found numerous applications not only in different parts of (traditional) mathematics but also in dynamical optimization, economics, physics etc [1].

We consider the spaces of the so called idempotent measures (Maslov measures) on compact Hausdorff spaces endowed with the weak topology. A measure m is a Maslov measure if m(A∪B)=max{m(A), m(B)} for all sets A, B. We identify the set of Maslov measures with the set of all functionals m: C(X)→R satisfying m(c)=c, m(j+c)=m(j)+c, and m(j⊕y)=m(j)⊕m(y).

The construction of the space of idempotent measures determines a functor on the category of compact Hausdorff spaces that shares some topological and categorical properties with the functor of probability measures. This functor is normal in the sense of E. Shchepin [2]. In particular, we show that, for this functor, a counterpart of the Kolmogorov theorem holds. We also show that the functor of idempotent measures preserves the class of open maps. The corresponding result for the probability measures is proved by Ditor and Eifler [3]. We also prove the existence of a counterpart of the Milyutin map for the spaces of idempotent measures.

Some geometric properties of the space of idempotent measures are considered. In particular, we prove that the space of idempotent measures of any infinite compact metrizable space is homeomorphic to the Hilbert cube. We also discuss the properties of the spaces of idempotent probability measures for nonmetrizable compacta.

It is known that the correspondence assigning to every probability measures on the factors the set of measures with given marginals on the product is continuous with respect to the Vietoris topology [4]. We demonstrate that this is not the case for the correspondence of idempotent measures on the product.

[1] V.P. Maslov, S.N. Samborskii, Idempotent analysis. - volume 13 of Advances in Soviet Mathematics. Amer. Math. Soc., Providence, 1992.

[2] E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi Mat. Nauk, 31 (1981), 3-62.

[3] S.Z. Ditor, L.Eifler, Some open mapping theorems for measures, Trans. Am. Math. Soc. 164(1972), 287-293.

[4] L. Eifler, Some open mapping theorems for marginals, Trans. Amer. Math. Soc. 211 (1975), 311-319.

Date received: May 9, 2008