Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $I\subseteq\mathbb R$ be an interval (of any form)
- $(\mathcal F_t)_{t\in I}$ be a filtration of $\mathcal A$
I've found two definitions of an $\mathcal F$-stopping time in some lecture books:
Definition 1: $\tau$ is called an $\mathcal F$-stopping time $:\Leftrightarrow$ $\tau:\Omega\to I\cup\left\{\infty\right\}$ is Borel-measurable and $$\left\{\tau\le t\right\}\in\mathcal F_t\;\;\;\text{for all }t\in I\;.\tag1$$
Definition 2: $\tau$ is called an $\mathcal F$-stopping time $:\Leftrightarrow$ $\tau:\Omega\to I\cup\sup I$ is Borel-measurable and $(1)$ is satisfied.
I'm curious whether one of these two definitions is somehow superior to the other one. For example, if $\mathcal M$ is some class of processes on $(\Omega,\mathcal A,\operatorname P)$ and we're using definition 1, we usually say that a function $M:\Omega\times I\to\mathbb R$ belongs locally to $\mathcal M$, if there is a sequence $(\tau_n)_{n\in\mathbb N}$ of $\mathcal F$-stopping times, called a localizing sequence for $M$, with $$\tau_n\xrightarrow{n\to\infty}\infty\tag2$$ and $$M^{\tau_n}\in\mathcal M\;\;\;\text{for all }n\in\mathbb N\;.\tag3$$ Note that by virtue of definition 1, $(2)$ actually means that $(\tau_n)_{n\in\mathbb N}$ eventually becomes constant. I wonder whether this might be a too restrictive assumption for the concept of a localizing sequence in general.
Using definition 2, we would replace $(2)$ by $$\tau_n\xrightarrow{n\to\infty}\sup I\;.\tag4$$
So, does it matter which definition we use and does one of them have crucial drawbacks?