On metrization of the idempotent measures functor and quantization dimensions

The paper continues the study of quantization dimensions of idempotent measures on an arbitrary metric compact space (X,ρ), started in [8]. The concept of an n-kernel of an idempotent measure, which plays an important role in calculating and estimating the metric ρI on the space of idempotent measures I(X), is defined. Formulas, that make the calculation of ρI(μ,ν),μ,ν∈I(X) convenient and geometrically clear, are obtained. It is shown that it is possible to get various metrizations of the idempotent measures functor I using the pseudometrics ρn introduced on I(X) in [2]. These metrizations have important general properties in terms of quantization dimensions. The existence of the metrization of the functor I, for which quantization dimensions are preserved in subspaces, is proved.