Sigma algebra proof On page 54 of these notes (section 3.3)
the following property of the sigma algebras is stated. 

Let:
  $H_t = 1_{\tau \leqslant t}$ (where $\tau$ and $H_t$ are random variables on some probability space, $1_A$ is indicator on A); $\mathcal{H}_t = \sigma(H_s : s \leq t)$ - the sigma algebra generated by all the $H_s$ for $s\leqslant t$,
  $\mathcal{F}$ is another sigma algebra.

Then for any set $A$ in $\sigma(\mathcal{F}, \mathcal{H}_t)$, $A \cap \{ \tau > t\} = F\cap \{ \tau > t\}$ for some $F$ in $\mathcal{F}$. 
This seems intuitive, but I'm struggling to give a really rigorous proof.

We can follow the following strategy: 


*

*the collection of subsets $A$ of $\Omega$ for which there exists $F\in\mathcal F$ such that $A\cap \{\tau>t\}=F\cap\{\tau>t\}$ is a $\sigma$-algebra.

*It contains both $\mathcal F$ and $\mathcal H_t$.
