Almost everywhere means at all points except possibly those in a set of measure zero. More precisely, let be a measure space. A property holds almost everywhere on if it holds on a measurable set for which .

For example, two measurable functions are equal almost everywhere if

Changing a function on a set of measure zero therefore does not change statements made only almost everywhere. The corresponding terminology for a probability measure is almost surely.

See also

Almost Everywhere Convergence, Almost Surely, Measure Zero

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References

Jeffreys, H. and Jeffreys, B. S. "'Measure Zero': 'Almost Everywhere.' " §1.1013 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 29-30, 1988.Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 1, 1991.

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Almost Everywhere

Cite this as:

Weisstein, Eric W. "Almost Everywhere." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlmostEverywhere.html

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