In measure theory (a branch of mathematical analysis), a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero (Halmos 1974). 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 sets 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).

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