  We study measures $\mu$ on the plane with two independent Alberti
representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such
measures are absolutely continuous with respect to Lebesgue measure. The
purpose of this paper is to quantify the result of A-C-P. Assuming that the
representations of $\mu$ are bounded from above, in a natural way to be defined
in the introduction, we prove that $\mu \in L^{2}$. If the representations are
also bounded from below, we show that $\mu$ satisfies a reverse H\"older
inequality with exponent $2$, and is consequently in $L^{2 + \epsilon}$ by
Gehring's lemma. A substantial part of the paper is also devoted to showing
that both results stated above are optimal.
