I am having a hard time trying to understand the following relation:
Consider that a stopping time is defined by $\{\tau \leq n \} \in \mathcal{F}_{n}$.
Now take two stopping times $\phi$ and $\tau$ such that $\{\phi \leq n \} \lt \{\tau \leq n \}$ with probability one, $\it{i.e.}$, $P(\phi \lt \tau) = 1 $.
Then it follows that $\{\tau \leq n \} \subseteq \{\phi \leq n \}$.
I do not understand the last statement. For me $\{\tau \leq n \}$ would have more elements of the probability space $\Omega$ than $\{\tau \leq n \}$ and, consequently, the relation would be the opposite.
Can someone please help me understand this? It can be a mathematical explanation or just an intuitive explanation! I would really appreciate some help.
Thanks in advance!
Supposet that $\phi<\tau$ in the sense that $\phi(\omega)<\tau(\omega)$ for every $\omega\in\Omega$ and let $\omega\in\Omega$ be fixed.
Then:$$\omega\in\{\tau\leq n\}\iff\tau(\omega)\leq n\implies\phi(\omega)\leq n\iff\omega\in\{\phi\leq n\}$$where $\implies$ is a consequence of $\phi(\omega)<\tau(\omega)$.
This is true for every $\omega\in\Omega$ so in that situation we can conclude that:$$\{\tau\leq n\}\subseteq\{\phi\leq n\}$$
Things are different if we only have $P(\phi<\tau)=1$ which is weaker than $\phi<\tau$.
Then some $\omega_0\in\Omega$ might exist with $\phi(\omega_0)>n\geq\tau(\omega_0)$ or equivalently: $$\omega_0\in\{\tau\leq n\}-\{\phi\leq n\}$$
So we cannot conclude in that case that $\{\tau\leq n\}$ is a subset of $\{\phi\leq n\}$.
What we can conclude in that situation is that $P(\{\tau\leq n\}-\{\phi\leq n\})=0$.