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Concept
Almost everywhere
Résumé
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory.
More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated.
The term almost everywhere is abbreviated a.e.; in older literature p.p. is used, to stand for the equivalent French language phrase presque partout.
A set with full measure is one whose complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space.
Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for almost all elements (though the term almost all can also have other meanings).
If is a measure space, a property is said to hold almost everywhere in if there exists a set with , and all have the property . Another common way of expressing the same thing is to say that "almost every point satisfies ", or that "for almost every , holds".
It is not required that the set has measure 0; it may not belong to . By the above definition, it is sufficient that be contained in some set that is measurable and has measure 0.
If property holds almost everywhere and implies property , then property holds almost everywhere. This follows from the monotonicity of measures.
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