On metrization of the functor of idempotent probability measures

Abstract

In idempotent mathematics, an analogue of a probability measure on a compactum X is a normed functional µ : C(X) → R, linear with respect to idempotent arithmetic operations. For ordinary probability measures, a meaningful theory of quantization has long been available, which has a wide range of applications (quantization of a measure is called its approximation by measures with finite supports). The question naturally arises of constructing a similar theory for idempotent probability measures. Quantization presupposes the presence of a metric on the space I(X) of idempotent probability measures, compatible with the topology and defining a metrization of the functor I of idempotent measures sensu V. V. Fedorchuk. A version of the metric on the space I(X) was defined in a joint paper by L. Bazilevich, D. Repovs, and M. Zarichnyi when proving that this space is homeomorphic to the Hilbert cube for any infinite metric compactum X. However, the structure of the metric of Bazilevich et al. is too complicated for it to be used for estimating approximations. In this paper, we propose a modified version of the metrization of the functor I, which is more convenient for constructing a theory of quantization of idempotent probability measures.