Measure-valued almost periodic functions and almost periodic selections of multivalued maps

This article contains a study of Stepanov almost periodic selections of multivalued maps $t\mapsto \operatorname {supp}\mu [\,\cdot\,;t]$, $t\in \mathbb R$. It is assumed that for a.e. $t\in \mathbb R$ the measure $\mu [\,\cdot\,;t]$ is a Radon probability measure on a complete metric space and $t\mapsto \operatorname {supp}\mu [\,\cdot\,;t]$, $t\in \mathbb R$, is a measure-valued almost periodic function.