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Almost everywhere

In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not is a subset of some null set. If the measure involved is complete, then the set of elements for which the property does not hold is a null set.

If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated. (The Lebesgue measure is complete.)

Occasionally, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements. The term almost all in addition has several other meanings however.

Here is a list of theorems that involve the therm "almost everywhere":

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