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

Question

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?

Answer 1

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