Prove that $m$ is ergodic.

Question

Let $X$ be a topological space, $f\colon X\rightarrow X$ be a function. Suppose that there exists a unique invariant Borel probability measure $m$. Prove that $m$ is ergodic.

Thank you.

Answer 1

So assume $m$ is not ergodic, so that there exists an invariant measurable set $A$ such that $$ 0<m(A)<1. $$

Now consider the following two measures $$ m_1(B):=\frac{m(A\cap B)}{m(A)}\quad\mbox{and}\quad m_2(B)=\frac{m(A^c\cap B)}{1-m(A)} $$ where $A^c=X\setminus A$.

Observe that these are two invariant Borel probability measures.

By assumption, they must be equal.

But $$ m_1(A)=1\neq 0=m_2(A). $$

So $m$ is ergodic.