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In this paper, we introduce the concept of almost-complete-closed time scales (ACCTS) that allows independent variables of functions to possess almost-periodicity under translations. For this new type of time scale, a class of piecewise functions with double-almost-periodicity is proposed and studied. Based on these, concepts of weighted pseudo-double-almost-periodic functions (WPDAP) in Banach spaces and a translation-almost-closed set are introduced. Further, we prove that the function space WPDAP0 affiliated to WPDAP is a translation-almost-closed set. Then, by introducing the concept of almost-uniform convergence for piecewise functions on ACCTS and using measure theory on time scales, some composition theorems of WPDAP and the completeness of the function space are proved.

Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales