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 $P$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: almost everywhere (AE)
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 10$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: almost everywhere (AE)