I'm having some trouble with the formatting here, so here's a screenshot of the problem we were given:
Let $\{ A_n \}_{n \geq 1}$ be a sequence of sets. Show that $ \lim\sup\limits_{n\to \infty} \mathbf{1}_{A_n} - \lim\inf\limits_{n\to \infty} \mathbf{1}_{A_n} = \mathbf{1}_{\{\lim\sup_{n \to \infty} A_n \backslash \lim\inf_{n \to \infty} A_n\}} $ and conclude that $A_n \to A$ if and only if $\mathbf {1}_{A_n}(\omega) \to \mathbf {1}_A(\omega)$ for all $\omega \in \Omega$. [$A \backslash B = A \cap B^c$.]
The problem is, that in the book from which the problem is taken from (Probability Essentials by Jean Jacod), we aren't given much help with understanding the notation. Earlier there is a proof where he defines $\lim \sup_{n \to \infty} A_n$ but I have no idea what $\lim \sup_{n\to \infty} \mathbf{1}_{\ldots}$ could mean.
Is this some common notation?
Thanks.