# Definition:Almost Everywhere

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## 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:

or:

This definition needs to be completed.In particular: Separate transcluded subpage for "almost surely", as it is frequently encountered and usedYou 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**.

## 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)**