About Gibbs measure

Question

I'm reading the book Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms by Rufus Bowen. After proving that the measure $\mu$ is $\sigma$-invariant (Lemma 1.13), where $\mu=h\nu$ and $h$ and $\nu$ satisfies the Ruelle's Perron Frobenius Theorem, the author defines the measure $\tilde{\mu}(f)=\displaystyle\lim_{n\to\infty}\mu((f\circ\sigma^{n})^*)$. Here we have that for $f\in\mathcal{C}(\Sigma_A)$ we can define $f^*\in \mathcal{C}(\Sigma_A^+)$ by $$f^*(\{x_i\}_{i=0}^{\infty})=\min\{f(\underline{y})\colon\underline{y}\in\Sigma_A, y_i=x_i, \text{for all } i\geq0\}.$$

My question is, why $\tilde{\mu}\in\mathcal{C}(\Sigma_A)^*$? In the book says that this is straightforward, but I don't see it.

Any help it will be appreciated.

Thanks in advance.

Answer 1

Two important properties:

$1$. Characteristic functions on cylinder sets are continuous functions.

$2$. Any continuous function can be approximated uniformly by linear combinations of them.

Actually, you can take at each step the linear combination coming from cylinders of a given length (all with the same length).

So all has to do with writing some triangle inequality and using this uniform approximation.