Isomorphism of probability spaces modulo null sets

An isomorphism modulo null sets between probability spaces $(X,\Sigma,\mu)$ and $(Y,\mathcal T,\nu)$ is a bijection $b:X_0\to Y_0$ between measurable subsets of measure one such that $b,b^{-1}$ are measurable and

\mu(b^{-1}E)=\nu(E)\qquad(E\subseteq Y_0\text{ measurable}).

Maps agreeing almost everywhere represent the same isomorphism.

What is preserved

Integration, measurable events modulo null sets, and $L^p$ norms are preserved. Topology, distance, and linear coordinates are additional structures that this definition need not preserve. Compatibility with dynamics is the extra requirement in measurable conjugacy.