Definition:Almost Everywhere

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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


 * $\left\{{x \in X: \neg P \left({x}\right)}\right\}$

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 $P$.