The uniform approximation of recurrent functions and almost recurrent functions

We consider the classes of functions $f\colon\mathbb R\to U$, taking values in a metric space $(U,\rho)$, which have Bochner transforms from the classes of recurrent functions and almost recurrent functions. We improve the preceding results on the uniform approximation of functions from classes under consideration by elementary functions from the same classes. These results can be applied to the investigation of the problem of the existence of almost recurrent selections for multivalued maps. The selections are supposed to satisfy a number of additional conditions. In the last section of the paper the variant of Lusin's theorem for recurrent functions is proved.