Q:

For an ergodic Invariant measure $\mu$, if a set $B$ has measure 0 can we imply $\mu(f^{-1}(B)) =0$?

We state $\mu$ is an ergodic invariant measure for $f$. We know that if a set, $A$ is invariant with respect to $f$ then $\mu(A) \in  \{0,1\}$. 
According to some lecture notes, we do not know set $B$ is invariant with respect to $f$ and that $\mu(B) = 0$. We are then told $\mu(f^{-j}(B)) = 0 , \ \forall j \in \mathbb{N}.$ How can we imply this is the case? 

A:

Ergodicty is defined for measure preserving transformations. The fact that $\mu (f^{-j} (B))=0$ comes from measure preserving nature of $f$, not ergodicity. 

