  We consider the relation between simplicial volume and two of its variants:
the stable integral simplicial volume and the integral foliated simplicial
volume. The definition of the latter depends on a choice of a measure
preserving action of the fundamental group on a probability space. We show that
integral foliated simplicial volume is monotone with respect to weak
containment of measure preserving actions and yields upper bounds on (integral)
homology growth. Using ergodic theory we prove that simplicial volume, integral
foliated simplicial volume and stable integral simplicial volume coincide for
closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable
residually finite fundamental group (being equal to zero in the latter case).
However, we show that integral foliated simplicial volume and the classical
simplicial volume do not coincide for hyperbolic manifolds of dimension at
least 4.
