Let $T$ be the one-sided shift on sequences with $k$ symbols. What are the sufficient conditions for $\mu_n \to \mu$ where $\mu_n$ are all invariant measures?
Is it sufficient to show that for each cylinder $A$, $\mu_n(A)\to\mu(A)$? I think yes, since I think characteristic functions of cylinders $\chi_A$ are dense in continuous functions, but I have not been able to prove this claim, or find anything with google searches.
I think the first observation to note is that cylinders in $[k]^{\mathbb{N}}$ are balls with respect to an ultra-metric of agreeing around the origin. This metric induces the standard product topology on the one-sided shift space.
Next note that the space is compact and so any continuous function $f:[k]^{\mathbb{N}}\to \mathbb{C}$ is uniformly continuous. So for every $\epsilon>0$ there exists $\delta>0$ such that $\omega, \eta \in [k]^{\mathbb{N}}$ with $d(\omega,\eta)<\delta$ implies that $\vert f(\omega)-f(\eta) \vert<\epsilon$.
There exists a length $\ell\in \mathbb{N}$, such that $\omega(j)=\eta(j)$ for all $1\leq j \leq \ell$ if and only if $d(\omega,\eta)<\delta$. Then consider all the indicators $\mathbf{1}_A$, where $A$ is any cylinder of length $\ell$.
Choosing for each cylinder $A$ an element $\omega_A\in A$, we define that $f_\epsilon:=\sum_{A} f(\omega_A)\cdot \mathbf{1}_A$. Then, $\Vert f-f_\epsilon\Vert_{\infty}<\epsilon$. So the linear span of indicator functions is dense in $C\big( [k]^{\mathbb{N}}, \mathbb{C} \big)$.
You can also possibly show this using some arguments relying on the $\pi-\lambda$ theorem.