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

Instead of almost everywhere, one may say:

Almost every element of $X$ has property $P$

or:

Almost all elements of $X$ have property $P$.

This definition needs to be completed. In particular: Separate transcluded subpage for "almost surely", as it is frequently encountered and used You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

In case that $\mu$ is a probability measure, one also says:

Property $P$ holds almost surely.

Also see

• Results about almost everywhere can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): 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): almost everywhere (AE)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): almost surely
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): almost surely