Need help in proving a unique invariant borel probability measure is ergodic.

Question

Let $X$ be a topological space and $f : X \rightarrow X$ be a function. Suppose that there exists a unique invariant borel probability measure m. We need show that m is ergodic : I assumed by contradiction that m is not ergodic. Then there is $A$ measurable such that $f^{-1}(A)=A$ and $0 < m(A) < 1$. How do I go from here?? Thank you for your help.