# Almost everywhere

From Example Problems

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* hold is a null set, i.e. is a set with measure zero. If used for properties of the real numbers, the Lebesgue measure is assumed unless otherwise stated.

Occasionally, instead of saying that a property holds almost everywhere, one also says that the property holds for **almost all** elements, though the term almost all also has other meanings.

Here are some theorems that involve the term "almost everywhere":

- If
*f*:**R**→**R**is a Lebesgue integrable function and*f*(*x*) ≥ 0 almost everywhere, then

- If
*f*: [*a*,*b*] →**R**is a monotonic function, then*f*is differentiable almost everywhere. - If
*f*:**R**→**R**is Lebesgue measurable and

- for every real numbers
*a*<*b*, then there exists a null set*E*(depending on*f*) such that, if*x*is not in*E*, the Lebesgue mean

- converges to
*f*(*x*) as decreases to zero. In other words, the Lebesgue mean of*f*converges to*f*almost everywhere. The set*E*is called the Lebesgue set of*f*.

- If
*f*(*x*,*y*) is Borel measurable on**R**^{2}then for almost every*x*, the function*y*→*f*(*x*,*y*) is Borel measurable.

In probability theory, the phrases become *almost surely*, *almost certain* or *almost always*, corresponding to a probability of 1.fr:Presque partout