information:classification, stylistic:tone
Almost sure equality

Two random variables X and Y are equal almost surely (denoted X = a.s. Y {\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}) if, and only if, the probability that they are different is zero:

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

where "ess sup" represents the essential supremum in the sense of measure theory.