I'm reading a book on Ergodic theory, and it says that given a set X with a sigma algebra A, and a measurable automorphism T, then you can take a set of ergodic measures $\mu_i$ $\epsilon$ M$_T(X)$ $i=1,...,n$ (where M$_T(X)$ is the set of T-invariant probability measures) $\mu_i \ll \mu_j$ i $\ne$ j, and define disjoint sets A$_i$ such that $\cup_{i=1}^n A_i$ = X and $\mu_j$($A_i$) = $\delta_{ij}$.
I guess I don't understand how you can partition the space such that each measure is independently supported on their respective $A_i$. While still having the same measure as the entire space.