in the lecture we formulated the ergodic theorem for an irreducible and positive recurrent transition matrix P with stationary distribution $\pi$, stating that for bounded functions $f$, it holds: $$\mathbb{P}\left[ \frac{1}{n}\sum_{k=0}^{n-1}f(X_k) \overset{n \rightarrow \infty}{\rightarrow} \sum_{i}f(i)\pi_i\right] =1$$
Furthermore, we had the convergence to equilibrium, stating that for a Markov Chain with initial distribution $\mu$, ergodic transition matrix P and the total variation distance, it holds: $$\lim_{n \rightarrow \infty}\|\mu P^n-\pi\|_{TV} = 0$$
I thought about how those to statements are related to each other and was wondering if I could follow the convergence to equilibrium from the Ergodic Theorem under the additional assumption of aperiodicity? And if so, is there an easy way to see this?
Thanks a lot for your help!