Almost surely

In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory. Although in many basic probability experiments there is no difference between almost surely and surely (that is, entirely certain to happen), the distinction is important in more complex cases relating to some sort of infinity. For instance, the term is often encountered in questions that involve infinite time, regularity properties or infinite-dimensional spaces such as function spaces. Basic examples of use include the law of large numbers (strong form) or continuity of Brownian paths.

The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of almost surely: an event that happens with probability zero happens almost never.

Formal definition

Let be a probability space. An event happens almost surely if . Equivalently, happens almost surely if the probability of not occurring is zero: . More generally, any event (not necessarily in ) happens almost surely if is contained in a null set: a subset of some such that . The notion of almost sureness depends on the probability measure . If it is necessary to emphasize this dependence, it is customary to say that the event occurs -almost surely or almost surely .