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 be a stopping time on the filtered probability space . Then the σ-algebra is called the σ-algebra of τ-past. Properties Monotonicity Is are two stopping times and almost surely, then Measurability A stopping time is always -measurable. Intuition The same way is all the information up to time , is all the information up time . The only difference is that 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 be the first time the random walk hit 10, would give you the information to answer that question. References Stochastic processes Families of sets