Ergodic measure

Question

Let $m$ be a $\phi$ Ergodic measure on a space $X$, where $\phi$ is a homeomorphism on $X$. Let $\mu_1$, $\mu_2$ be two $\phi$-invariant $\sigma$ finite measures on $X$ such that $m$ is equivalent to both $\mu_1$ and $\mu_2$. I want to prove that $\mu_1$, $\mu_2$ are scalar multiplies of each other. By Radon Nikodym theorem there is a $\mu_2$ measurable function $f$ such that $\mu_1(B)=\int fd\mu_2$ for all Borel sets $B$. I proved that $f=f\circ \phi$, $\mu_2$ almost everywhere. How ergodicity of $m$ will imply that $f$ is a constant?

Answer 1

Hints: For any Borel set $A$ we have $\mu_2(f^{-1}(A))\in \{0,1\}$ becaue $f^{-1}(A)=\phi^{-1}(f^{-1}(A))$. Let $c=\sup \{x: f^{-1}(-\infty, x)=0\}$. Show that $f=c$ a .e. $[\mu _2]$.