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":

• 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.