Note on Besicovitch's Theorem on the Possible Values of Upper and Lower Derivatives

Let $B_1,\dots,B_k$ be Busemann–Feller and regular differential bases composed of intervals of the corresponding dimensions. It is proved that if $B_1,\dots,B_k$ satisfy a certain condition (called the completeness condition), then, for their Cartesian product $B_1\times \dotsb\times B_k$, an analog of Besicovitch's theorem on the possible values of strong upper and lower derivatives is valid.