There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.

My question is: are any such measures also $T^{-1}$-invariant? BYO sigma-algebra.

The question is trivial if we consider a two-sided shift space. But for the one-sided space I can only think of a family of atomic examples:

- $m_1(x) = \delta_0(x)$ with $\sigma$-algebra $\mathcal B = \left\{\emptyset,\left\{0\right\},X\right\}$, which assigns all the mass to the point $0$,
- $m_3$ with $\sigma$-algebra $\mathcal B = \left\{\emptyset,\left\{\frac13\right\},\left\{\frac23\right\},\left\{\frac13,\frac23\right\},X\right\}$, which assigns half mass to $\frac13 = 0.0101010101\cdots$ and the other half to $\frac23 = 0.10101010\cdots$, or
- any measure $m_k$ with $k\geq2$ and $\gcd(k,2)=1$. Where $\mathcal B$ is defined as the smallest $\sigma$-algebra containing $\left\{\frac{j}k : 0 \lt j \lt k\right\}$ and $m_k$ assigns equal mass to all the points: $m_k\left(\frac{j}{k}\right) = \frac{1}{k-1}, 0 < j < k$.

After a bit more thought, the sigma-algebra for such a measure can not contain finite cylinders of measure less than one. For example if $[0]_n = \left\{ x \in X : x_n = 0\right\}$ then $T^n[0]_n = X$ and by $T^{-1}$ invariance of the measure $\mu([0]_n) = \mu(X) = 1$.

This question was posted earlier on [https://math.stackexchange.com/q/1769011/154377].