I'm a worker and I'm studying Stochasstic processes from the book Stochastic Calculus and Application by Cohen and Elliott.
In particular, I am reading the Chapter 4 on Discrete Time Stochastic Processs, that is, processes of the form $X=\{X_n \}_{n \in\mathbb{T}}$ where $\mathbb{T}$ could be: $$ \mathbb{T}=\mathbb{Z}^+ \;\;\; \text{or}\;\;\;\mathbb{T}= \overline{\mathbb{Z}}^+= \mathbb{Z}^+\cup\{\infty\} $$ Before starting the section on Martingale Convergence the author says, consider that there is a proability space $(\Omega,\mathcal{F}, P)$ equipped with a filtration $\{\mathcal{F}_n\}_{n\in \mathbb{T}}$:
For the sake of notational simplicity, if our setting is $\mathbb{T}=\mathbb{Z}^+$, then we define $\mathcal{F}_{\infty}=\bigvee_n \mathcal{F}_n$ , and so have a filtration defined for $\mathbb{T}= \overline{\mathbb{Z}}^+$. We write $\mathcal{F}_{\infty^-}=\bigvee_{n<\infty} \mathcal{F}_n$ and so know $\mathcal{F}_{\infty^-} \subseteq \mathcal{F}_{\infty}$.
Now the $\bigvee_n \mathcal{F}_n$ is defined before as the $\sigma$ algebra generated by filtration. i.e. $\bigvee_n \mathcal{F}_n=\sigma(\cup_n\mathcal{F_n})$.
I do not understand the definition of $\mathcal{F}_{\infty^-}$; what is the difference between $\mathcal{F}_{\infty^-}$ and $\mathcal{F}_{\infty}$?
Note that I have written exactly the same words of the book; any mistakes or ambiguities are not my fault, I am asking precisely because I think that this notation in ambiguous. Have you ever find this notation in other authors?
edit: to give some additional info; After this "definition" there is a theorem:
Suppose that $\{X_n\}_{n \in\mathbb{Z}^+}$ is a super-martingale such that $\sup_n E[X_n]<\infty$, then the sequence $\{X_n\}_{n \in\mathbb{Z}^+}$ converges almost surely to an integrable random variable $X_{\infty} \in L^1(\mathcal{F}_{\infty -})$.
Onestly I do not get the point. It seems to me that this notation is not used in the majority of cases.