Approximation and transfer of properties between net-type and translation invariant convex differentiation bases

Let $\mathbf B$ be a net-type convex differentation basis and $\mathbf H$ be a translation invariant convex differentation basis. We show the transfer of differentiation properties for non-negative functions from $\mathbf B$ to $\mathbf H$ provided $\mathbf B$ is a subbasis of some translation invariant density basis and each set $H$ forming $\mathbf H$ can be approximated by some set $B$ forming $\mathbf B$ in the sense that $| B\triangle H|\leq c | B\cap H|$, where $c$ is a positive constant not depending on $H$. Some applications for net-type bases formed by rectangles with dyadic type constraints are given.