Almost everywhere
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. In the subject of probability, which is largely based in measure theory, the notion is referred to as almost surely.
More specifically, a property holds almost everywhere if the set of elements for which the property does not hold is a set of measure zero (Halmos 1974), 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 is contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is 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, which 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 also has other meanings).
If is a measure space, a quality P is said to hold almost everywhere in X if μ({x∈X: ¬P(x)}) = 0. Another common way of expressing the same thing is to say that "almost every point satisfies P" or "for almost every x, P(x) holds".
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