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

Invariant Measures with Bounded Variation Densities for Piecewise Area Preserving Maps