Equivalent Conditions for being a Sigma Field up to a Stopping Time

Last updated Nov 1, 2022

Let ${\mathcal{B}_{n} : n \in \mathbb{N}}$ be a Discrete-Time Filtration on $\Omega$ and let $\nu : \Omega \to \bar{\mathbb{N}}$. Then the following Sigma Algebras are equal:

1. $\mathcal{B}_{\nu}$
2. ${B \in \mathcal{B}{\infty}: B \cap [\nu \leq n] \in \mathcal{B}{n}}$

TODO. Use Equivalent Conditions for being a Stopping Time

1. It is not immediately clear that (2) is a Sigma Algebra, but we know (1) is, so it suffices to show equivalence.
2. It is not in general true that if $B \in \mathcal{B}\nu$, then $B \cap [\nu > n] \in B{n}$. TODO what is an example of this? However, if $A \cap [\nu > n] \in \mathcal{B}{n}$ for all $n \in \mathbb{N}$, then $(A^{C} \cap [\nu > n]) \sqcup (A \cap [\nu \leq n]) \in \mathcal{B}{n}$, and $A^{C} \cap [\nu > n] \in \mathcal{B}{n}$, so $A \cap [\nu \leq n] \in \mathcal{B}{n}$. Thus $A \in \mathcal{B}{\nu}$ and ${A : A \cap [\nu > n] \in \mathcal{B}{n}} \subset \mathcal{B}_\nu$.