Metrical Bochner criterion and metrical Stepanov almost periodicity

We analyze the metrical Bochner criterion and a new class of multi-dimensional metrically Stepanov almost periodic type functions. We clarify the main structural properties for the introduced classes of functions, including the Bochner criterion, and provide certain applications to Doss-$p$-almost periodic functions. We also study the extensions of almost periodic sequences and briefly explain how we can apply the established theoretical results to the abstract Volterra integro-differential equations.