  We investigate the properties of absolutely continuous invariant probability
measures (ACIPs), especially those measures with bounded variation densities,
for piecewise area preserving maps (PAPs) on $\mathbb{R}^d$. This class of maps
unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where
Lebesgue measure is locally preserved. Using a functional analytic approach, we
first explore the relationship between topological transitivity and uniqueness
of ACIPs, and then give an approach to construct invariant measures with
bounded variation densities for PWIs. Our results "partially" answer one of the
fundamental questions posed in \cite{Goetz03} - to determine all invariant
non-atomic probability Borel measures in piecewise rotations. When restricting
PAPs to interval exchange transformations (IETs), our results imply that for
non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very
irregular densities, i.e., they have unbounded variation.
