Definition:Almost Everywhere

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

A property $\map P x$ of elements of $X$ is said to hold ($\mu$-)almost everywhere if the set:


 * $\set {x \in X: \neg \map P x}$

of elements of $X$ such that $P$ does not hold is contained in a $\mu$-null set.

Also known as
Alternatively, one may say:
 * Almost every element of $X$ has property $P$

or:
 * Almost all elements of $X$ have property $P$.

In case that $\mu$ is a probability measure, one also says:
 * Property $P$ holds almost surely.