ISSN:
1573-8876
Keywords:
almost periodic functions
;
probability measures
;
Stepanov almost periodicity
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$Mod\mu = \sum\limits_{\mathcal{F} \in C_b (\mathcal{U},\mathbb{R})} {Mod\int_u {\mathcal{F}(x)\mu [dx; \cdot ]} .} $$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02355007
Permalink