My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim\_{n\to\inf}||P(S\_n\in\cdot)-P(S\_n+k\in\cdot)||\_{tv}=0$$
This is from coupling lectures in probability theory.



"X \in \cdot" in Probability Measure
If $S\_n$ is a real-valued random variable, then I would read $P(S\_n \in \cdot)$ as denoting the probability measure $P \circ S\_n^{-1}$ on $\mathbb{R}$, i.e. the set function $B \mapsto P(S\_n \in B)$ where $B \in \mathcal{B}\\_{\mathbb{R}}$, the Borel $\sigma$-algebra on $\mathbb{R}$. The quantity in norm bars is then the total variation distance between these two measures, which means the given expression can be rewritten
$$
\lim\_{n \to \infty} \sup\_{B \in \mathcal{B}\_\mathbb{R}} |P(S\_n \in B) - P(S\_n + k \in B)|
$$


