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.
closed as too localized by Wadim Zudilin, José FigueroaO'Farrill, Yemon Choi, Andres Caicedo, Martin Brandenburg Jan 9 '11 at 9:59This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


If $S_n$ is a realvalued 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 $$ \lim_{n \to \infty} \sup_{B \in \mathcal{B}_\mathbb{R}} P(S_n \in B)  P(S_n + k \in B) $$ 


I think the answer is that the $\cdot$ induces an anonymous function, e.g. the above is equal to: $$\lim_{n\to\inf}f_{tv}=0$$ with f a function on the appropriate domain having $$f(x) := P(S_n=x)P(S_n+k=x)$$ 

