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Almost everywhere

The function 1/x is differentiable and continuous almost everywhere, more precisely, everywhere except at x = 0.

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,[1][2] 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.;[3] in older literature p.p. is used, to stand for the equivalent French language phrase presque partout.[4]

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).

  1. ^ Weisstein, Eric W. "Almost Everywhere". mathworld.wolfram.com. Retrieved 2019-11-19.
  2. ^ Halmos, Paul R. (1974). Measure theory. New York: Springer-Verlag. ISBN 0-387-90088-8.
  3. ^ "Definition of almost everywhere | Dictionary.com". www.dictionary.com. Retrieved 2019-11-19.
  4. ^ Ursell, H. D. (1932-01-01). "On the Convergence Almost Everywhere of Rademacher's Series and of the Bochnerfejér Sums of a Function almost Periodic in the Sense of Stepanoff". Proceedings of the London Mathematical Society. s2-33 (1): 457–466. doi:10.1112/plms/s2-33.1.457. ISSN 0024-6115.

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