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 term "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*) d*x*≥ 0. - If
*f*: [*a*,*b*] →**R**is a monotonic function, then*f*is differentiable almost everywhere.