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 .

Elvis Presley

Elvis Aaron Presley (January 8, 1935 – August 16, 1977) was an American singer and actor. Regarded as one of the most significant cultural icons of the 20th century, he is often referred to as "the King of Rock and Roll", or simply, "the King".

Presley was born in Tupelo, Mississippi, as a twinless twin, and when he was 13 years old, he and his family relocated to Memphis, Tennessee. His music career began there in 1954, when he recorded a song with producer Sam Phillips at Sun Records. Accompanied by guitarist Scotty Moore and bassist Bill Black, Presley was an early popularizer of rockabilly, an uptempo, backbeat-driven fusion of country music and rhythm and blues. RCA Victor acquired his contract in a deal arranged by Colonel Tom Parker, who managed the singer for more than two decades. Presley's first RCA single, "Heartbreak Hotel", was released in January 1956 and became a number-one hit in the United States. He was regarded as the leading figure of rock and roll after a series of successful network television appearances and chart-topping records. His energized interpretations of songs and sexually provocative performance style, combined with a singularly potent mix of influences across color lines that coincided with the dawn of the Civil Rights Movement, made him enormously popular—and controversial.