Note on Besicovitch’s theorem on the possible values of upper and lower derivatives

LetB 1, ..., B k be Busemann-Feller and regular differential bases composed of intervals of the corresponding dimensions. It is proved that if B 1, ...,B k satisfy a certain condition (called the completeness condition), then, for their Cartesian product B 1 × ... × B k , an analog of Besicovitch’s theorem on the possible values of strong upper and lower derivatives is valid.