Almost periodic measure-valued functions

Weakly almost periodic measure-valued functions $\mathbb R\ni t\to\mu[\,\cdot\,;t]$ taking values in the space $\mathscr M(U)$ of Borel measures of variable sign in a complete separable metric space $U$ are considered. A norm ${\|\cdot\|}_w$ introduced in the space $\mathscr M(U)$ defines a metric on the set of probability Borel measures that is equivalent to the Levy–Prokhorov metric. A connection between the almost periodicity of a measure-valued function $t\to\mu[\,\cdot\,;t]\in (\mathscr M(U),{\|\cdot\|}_w)$ and its weak almost periodicity (both in the sense of Bohr and in the sense of Stepanov) is established.