  Let $Q=(0,1)^2$ be the unit square in $\mathbb{R}^2$. We prove that in a
suitable complete metric space of $BV$ homeomorphisms $f:Q\rightarrow Q$ with
$f_{|\partial Q}=Id$, the generical homeomorphism (in the sense of Baire
categories) maps a null set in a set of full measure and vice versa. Moreover
we observe that, for $1\leq p<2$, in the most reasonable complete metric space
for such problem, the family of $W^{1,p}$ homemomorphisms satisfying the above
property is of first category, instead.
