Definition:Almost Everywhere

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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 $P$


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

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

Property $P$ holds almost surely.