Sigma-Algebra of Events Prior to a Stopping Time

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be a real number, let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$, and let $\tau$ be a stopping time of $(\mathcal{F}_t)_{t\in[0,T]}$.

The $\sigma$-algebra of events prior to $\tau$ is the family

$$\mathcal{F}_\tau=\big\{A\in\mathcal{F}_T:\ A\cap\{\tau\le t\}\in\mathcal{F}_t\ \text{for every }t\in[0,T]\big\},$$

where $\{\tau\le t\}$ denotes the set $\{\omega\in\Omega:\tau(\omega)\le t\}$.