# Confusion regarding stopping times

I am having a hard time with “stopping time” definitions, notation and reasoning behind it.

τ∈{0,1,2,…;+∞}\tau \in \{0,1,2,…;+\infty\}

What formaly does {τ≤n}\{ \tau \leq n \} mean? Does it represent an event? Particularly in {τ≤n}={w:τ(w)≤n}∈Fn \{ \tau \leq n \} = \{ w:\tau(w) \leq n \} \in \mathscr{F_n} . What is τ(w)\tau(w) then?
Can you please explain meaning of the {τ=n}={τ≤n}∖{τ≤n−1}∈Fn
\{ \tau = n \} =
\{ \tau \leq n \} \setminus \{ \tau \leq n – 1 \} \in \mathscr{F_n}
? I unfortunately have no intuitive understandring of this statement.
Can you please refer to clear construction examples of sigma algebra / filtration needed for stopping time to be measurable / adapted.
Can you please recommend good book/article with clear and intuituve guidance to grasp a motivation and understanding of stopping times?

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