Measure-valued almost periodic functions

Abstract

We consider Stepanov almost periodic functions μ ∈$\mathcal{S}(\mathbb{R},{\text{ }}\mathcal{M})$ ranging in the metric space$\mathcal{M}$ of Borel probability measures on a complete separable metric space$\mathcal{U}{\text{ }}(\mathcal{M}$ is equipped with the Prokhorov metric). The main result is as follows: a function$t \to \mu \left[ { \cdot ;t} \right] \in \mathcal{M},{\text{ }}t \in \mathbb{R}$, belongs to$\mathcal{S}(\mathbb{R},{\text{ }}\mathcal{M})$ if and only if for each bounded continuous function$\mathcal{F} \in C_b (\mathcal{U},\mathbb{R})$, the function$\int_u {\mathcal{F}(x)\mu [dx; \cdot ]}$ is Stepanov almost periodic (of order 1) and

$$Mod\mu = \sum\limits_{\mathcal{F} \in C_b (\mathcal{U},\mathbb{R})} {Mod\int_u {\mathcal{F}(x)\mu [dx; \cdot ]} .} $$