What is the meaning, both intuitively and mathematically, behind the probability of a set?

Question

I'm currently studying a course on applied probability and I keep running into a notation that I can't quite understand.

Let's say we are working in a progrability space $(\Omega,\mathcal{F},P)$ and we have some sequence of independent indentically distributed variables $\{X_{i}:i \in \mathbb{N} \}$ with each $X_{i}$ a martingale. We have a stopping time $\tau$. I realise that in order to make any sense of this setting I need to give additional information about the problem but I'd like to just focus on the meaning of the following statements:

1. $\mathbb{P}(\{\tau = \infty \}) < 1$
2. $\forall n\in\mathbb{N}:\mathbb{P}(\{\tau \geq n\}) \geq \mathbb{P}(\{\tau = \infty \})$

I guess in words they would mean: "The probability that our stopping time equals infinity is smaller than $1$". However, what do the statements mean with regard to our probability space? Does the first statement mean that for any arbitrary $\omega$, the probability of $\tau(\omega)$ not being equal to $\infty$ is larger than $0$? The second statement is already considerably harder to wrap my head around than the first one.

I'm quite struggling with this relatively elementary notation so I hope I don't get laughed off this board, but any help is appreciated. The notation becomes especially hard to deal with when random walks and stopping times for those types of processes get introduced.

Answer 1

1) The statement $P[\tau = \infty] < 1$ means that if we randomly select an outcome $\omega$ to produce $\tau(\omega)$, the probability that our $\tau(\omega)$ is equal to infinity is less than 1. Indeed it implies that the probability that our $\tau(\omega)$ is less than infinity is larger than 0.

2) The second statement means that if we compare the probability that $\tau(\omega)=\infty$ to the probability that $\tau(\omega) \geq n$, the first is less than or equal to the second. This is because if $\tau(\omega)=\infty$ then we know that $\tau(\omega) \geq n$ must also be true. In particular: $$ \{\tau(\omega) = \infty\} \subseteq \{\tau(\omega) \geq n\}$$ and recall $A\subseteq B \implies P[A] \leq P[B]$.