  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.
