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 doesn't is a subset of some null set. If the measure involved is complete, then the set of elements for which the property doesn't 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":
- A bounded function f : [a, b] → R is Riemann integrable if and only if it is continuous almost everywhere.
- If f : R → R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then ∫ f(x) dx ≥ 0.
- If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere.
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