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.