Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$:
$T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$.
There are lots of examples of $T$-invariant measures. For example the Bernoulli $\left(\frac{1}{2},\frac{1}{2}\right)$ measure, which assigns mass $2^{-n}$ to cylinders of length $n$.
My question is: are there ergodic, non-atomic measures on this space which are both $T$-invariant and $T^{-1}$-invariant? BYO sigma-algebra.
The question is trivial if we consider a two-sided space. But for a one-sided space I can think of atomic examples, such as
- $m_1(x) = \delta_0(x)$ 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 $\frac13 = 0101010101\cdots$ and the other half to $\frac23 = 10101010\cdots$, or
- any measure $m_k$ with $k\geq2$ and $\gcd(k,2)=1$ (so the binary expansion of $\frac{1}{k}$ is infinite and repeating). Where $\mathcal B$ is defined as the smallest $\sigma$-algebra containing $\left\{\frac{j}k : 0 \lt j \lt k\right\}$ and $m_k\left(\frac{j}{k}\right) = \frac{1}{k-1}, 0 < j < k$.
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 $[b]_n = \left\{ x \in X : x_n = b\right\}$ then $T^n[b]_n = X$ and by $T^{-1}$ invariance of the measure $\mu([b]_n) = \mu(X) = 1$.
