On Busemann–Feller extensions of translation invariant density differentiation bases

Abstract

Abstract It is known that a translation invariant density differentiation basis B formed of convex sets and its Busemann–Feller extension B BF {B_{\operatorname{BF}}} have similar properties, in particular, the Busemann–Feller extension differentiates the same class of non-negative functions as the basis B. We prove the optimality of the convexity condition in this result.