Almost everywhere

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In measure theory (a branch of mathematical analysis), a property holds almost everywhere (sometimes abbreviated as a.e.) if the set for which the property does not hold is negligible in a technical sense. 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 phrase presque partout. A set with full measure is one whose complement is of measure zero. In...

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