What does the following notation mean: $\tau = \inf \{n\in \mathbb{N} \mid X_n \in B \} $

2017-12-14 05:20:38

Let $(\Omega, \mathbb{P})$ be a probability space and $B\subseteq \Omega$ an event. Let further be $X_1,...,X_n$ random variables.

What does $\tau = \inf \{n\in \mathbb{N} \mid X_n \in B \} $ mean?

For usual the notation $X\in M$ for a set $M$ means $\{w\in \Omega\mid X_n(w) \in M\}$, but that requires that $M$ is part of the image of $X_n(\Omega)$, i.e. $M \subseteq X_n(\Omega)$.

So, for $B$ that notation probably means something different, but what?