On the differentiation of integrals with respect to translation invariant convex density bases

Abstract

For a translation invariant convex density basis $B$ it is shown that its Busemann-Feller extension $B_{\mathrm{BF}}$ has close to $B$ properties, in particular, $B_{\mathrm{BF}}$ differentiates the same class of non-negative functions as $B$. Using the similarity between properties of the bases $B$ and $B_{\mathrm{BF}}$ some results known for Busemann-Feller bases are transferred to bases without restriction of being Busemann-Feller.