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 9:59
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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 $$f(x) := P(S_n=x)P(S_n+k=x)$$ 

