For the circle-doubling map $T:[0,1]\to [0,1]$, $x\mapsto 2x \mod(1)$. I want to show the ergodic probability measures are dense in the space of invariant probability measures.
My attempt: I know the Lebesgue measure $m$ is ergodic. Using binary expansion, the measure-preserving system $(T,m)$ is isomorphic to the Bernoulli shift $(1/2, 1/2)$. Using the same idea, for any $p\in (0,1)$, we can put the product measure of $(p,1-p)$ on $\{0,1\}^\mathbb{N_0}$, then pull it back to $[0,1]$ to get ergodic measures. But I am not sure how to prove the density.
Thank you in advance.