On Weyl almost periodic measure-valued functions

We consider measure-valued functions ${\mathbb{R}}\ni t\to \mu [.;t]$ taking values in the metric space $({\mathcal M}_0(U),\rho _w)$ of probability Borel measures defined on the $\sigma$-algebra of Borel subsets of a complete seperable metric space $U$. The metric space $({\mathcal M}_0(U), \rho _w)$ is endowed with the metric $\rho _w$ equivalent to the Lévy–Prokhorov metric. It is proved that the measure-valued function ${\mathbb{R}}\ni t\to \mu\, [\,.\,;t]\in ({\mathcal M}_0(U),\rho _w)$ is Weyl almost periodic if and only if the functions $\int\limits_U{\mathcal F}(x)\, \mu\, [\,dx;\,.\,]$ are Weyl almost periodic (of order 1) for all bounded continuous functions ${\mathcal F}:U\to {\mathbb{R}}$.