Showing a measure is transformation invariant as in the proof of Krylov Bogolubov

Question

First, context: Let $X$ be a compact metric space, and $C(X)$ the set of continuous functions on $X$ equipped with the sup norm. Let $T:X\to X$ continuous. Fix $x \in X$. Then one can show that $S_f^n(x) =\displaystyle \frac1n\sum_{k=0}^{n-1}f(T^k(x))$ is a bounded linear functional on $C(X)$, which has a convergent subsequence for all $g\in C(X)$. Denote the limit to be $S_g^\infty(x)$. This gives rise to the linear functional $L_x(g) = S_g^\infty(x)$, which is positive, so by Riesz Representation Theorem, we have $L_x(g) = \int_X gd\mu$ for some Borel probability measure.

Now we are basically at my question. The author of the book I'm using (Dynamical Systems by Brin & Stuck) next shows that $S_g^\infty (x) = S_g^\infty (Tx)$ to conclude that $\mu$ is $T$-invariant. However, I don't know why this is.

If we could work with characteristic functions, we would have the following:

$\mu(A) = \int_X {\mathcal{X}}_A d\mu = L_x(\mathcal{X}_A)= L_x(\mathcal{X}_A\circ T) = \int_X \mathcal{X}_A \circ T d\mu = \mu(T^{-1}(A)).$

However, we've been working with continuous functions. It is not true, I don't think, that we can approximate simple functions by continuous functions in the sup norm. So I'm quite confused about how we get that $\mu$ is $T-$invariant. Moreover, I'm not sure I get why $T$ being continuous is necessary.

For more context, here is the statement of the the theorem.

Let $X$ be a compact metric space and $T: X → X$ a continuous map. Then there is a $T$-invariant Borel probability measure $\mu$ on $X$.

Answer 1

It follows from looking at the averages that $$S^\infty_g(Tx)=\int_X(g\circ T)\,d\mu.$$ So $$\int_X g\,d\mu=\int_X(g\circ T)\,d\mu$$ for all $g$ continuous. Since continuous functions are dense in $L^1(X,\mu)$ (or apply Lusin theorem and then Tietze theorem), you can simply replace $g$ by a characteristic function.

Answer 2

Maniulating the finite version of the equality of interest into a difference and expanding the definitions yields much cancellation. (The sequence of points you evaluate at in one expression is almost the same as in the other, except for one endpoint of each.) \begin{align*} S_g^{n}(Tx) - S_g^{n}(x) &= \frac{1}{n} \left( \sum_{k=0}^{n-1} g(T^k Tx) - \sum_{k=0}^{n-1} g(T^k x) \right) \\ &= \frac{1}{n} \left( \sum_{k=0}^{n-1} g(T^{k+1} x) - \sum_{k=0}^{n-1} g(T^k x) \right) \\ &= \frac{1}{n} \left( \sum_{k=1}^{n} g(T^k x) - \sum_{k=0}^{n-1} g(T^k x) \right) \\ &= \frac{1}{n} \left( g(T^n x) - g(T^0 x) \right) \end{align*} Since $g$ is bounded, the expression in parentheses is bounded, so in the limit as $n \rightarrow \infty$, this last expression is zero.