of view of the field with one element, an ordering on a set corresponds to a maximal flag (a filtration on a vector space), considering a set to be a vector space over the field with one element. Measure theory In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of -algebras on a measurable space. That is, given a measurable space , a filtration is a sequence of -algebras with where each is a non-negative real number and The exact range of the "times" will usually depend on context: the set of values for might be discrete or continuous, bounded or unbounded. For example, Similarly, a filtered probability space (also known as a stochastic basis) , is a probability space equipped with the filtration of its -algebra . A filtered probability space is said to satisfy the usual conditions if it is complete (i.e., contains all -null sets) and right-continuous (i.e. for all times ). It is also useful (in the case of an unbounded index set) to define as the -algebra generated by the infinite union of the 's, which is contained in : A σ-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated, or "questions that can be answered at time ". Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information. A typical example is in mathematical finance, where a filtration represents the information available up to and including each time , and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available. Relation to stopping times: stopping time sigma-algebras Let be a filtered probability space. A random variable is a stopping time with respect to the filtration , if for all . The stopping time -algebra is now defined as . It is not difficult to show that is indeed a -algebra. The set encodes information up to the random time in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about it from arbitrarily often repeating the experiment until the random time is .