Measure-valued almost periodic functions

L. I. Danilov

Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 57-68

Résumé

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