Σ-Algebra of τ-past

The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probability theory.

Definition
Let $$ \tau $$ be a stopping time on the filtered probability space $$ (\Omega, \mathcal A, (\mathcal F_t)_{t \in T}, P ) $$. Then the σ-algebra
 * $$ \mathcal F_\tau:= \{ A \in \mathcal A \mid \forall t \in T \colon \{ \tau \leq t \} \cap A \in \mathcal F_t\} $$

is called the σ-algebra of τ-past.

Monotonicity
Is $$ \sigma, \tau $$ are two stopping times and
 * $$ \sigma \leq \tau $$

almost surely, then
 * $$ \mathcal F_\sigma \subset \mathcal F_\tau. $$

Measurability
A stopping time $$ \tau $$ is always $$ \mathcal F_\tau$$-measurable.

Intuition
The same way $$\mathcal{F}_t$$ is all the information up to time $$t$$, $$\mathcal{F}_\tau$$ is all the information up time $$\tau$$. The only difference is that $$\tau$$ is random. For example, if you had a random walk, and you wanted to ask, “How many times did the random walk hit −5 before it first hit 10?”, then letting $$\tau$$ be the first time the random walk hit 10, $$\mathcal{F}_\tau$$ would give you the information to answer that question.