For an arbitrary Markov chain, I'm trying to show that $\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert}\to 0$ iff $\sum_j \lvert{ P_{ij}^n - \nu(j)\rvert}\to 0$, where $\theta_n^{-1}$ is a shift operator taking the sequence $X_0, X_1,\ldots$ to $X_n, X_{n+1},\ldots$
I think this has something to do with the following identity $$\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert} = \frac{1}{2} \sum_{i,j} \mu(i)\lvert{ P_{ij}^n - \nu(j)\rvert}$$
I just don't see why the RHS goes to $0$ iff $\lim_{n\to \infty} \sum_j \lvert{ P_{ij}^n - \nu(j) \rvert} \to 0$ for all $i$.
**Here, $\mu$ and $\nu$ are both probability measures.