# Lebesgue-continuous Borel probability measures on $[0, 1)$ ergodic with respect to the doubling map $Tx = 2x \mod 1$

I'm learning about unique ergodicity and how a transformation can have different measures with respect to which it's ergodic. I read that, for example, if $$T : [0, 1) \to [0, 1)$$ is the doubling map $$Tx = 2 x \mod 1$$, then there are many Borel probability measures on $$I$$ which make $$T$$ ergodic. For example, if I choose any $$p \in [0, 1]$$, then I can define a Borel probability measure $$\mu_p$$ on the binary cylinders which amounts essentially to flipping a coin with probability $$p$$ of landing Heads, where Lebesgue measure is $$\mu_{1 / 2}$$.

However, I noticed that different values of $$p$$ gave singular measures, i.e. $$p \neq q \Rightarrow \mu_p \perp \mu_q$$. My questions is if there exist any Lebesgue-continuous Borel probability measures on $$[0, 1)$$ which $$T$$ is ergodic with respect to. Are different ergodic measures necessarily singular? Is there some more general theorem about this? Or am I missing some more obvious Lebesgue-continuous ergodic measure for this transformation?

## 1 Answer

After thinking on it, I think that these different ergodic measures will necessarily be singular. Basically, if $$\mu, \nu$$ are distinct ergodic measures, then there's some Borel $$A \subseteq I$$ such that $$\mu(A) = \nu(A)$$. Then we know that \begin{align*} \mu \left( \left\{ x \in I : \lim_{k \to \infty} \frac{1}{k} \sum_{j = 1}^k \chi_A \left( T^{j - 1}(x) \right) = \mu(A) \right\} \right) & = 1, \\ \nu \left( \left\{ x \in I : \lim_{k \to \infty} \frac{1}{k} \sum_{j = 1}^k \chi_A \left( T^{j - 1}(x) \right) = \nu(A) \right\} \right) & = 1 . \end{align*} But these two sets must necessarily be disjoint.

• mark as accepted by clicking check mark if this is right. if ur unsure whether this is right, let me know and ill read question and ur answer – mathworker21 Apr 16 '20 at 0:01