  We study measure-theoretical aspects of torus piecewise isometries. Not much
is known about this type of dynamical systems, except for the special case of
one-dimensional interval exchange mappings. The last case is fundamentally
different from the general situation in the presence of an invariant measure
(Lebesgue measure), which helps a lot in the analysis. Due to the absence of
good methods of analysis of general systems with discontinuities, even the
existence of invariant measures of the torus piecewise isometries was an open
question. We establish sufficient conditions for the existence/absence of
invariant measures for this class of systems. Technically, our results are
based on the approximation of the maps under study by weakly periodic ones.
