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:
- 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)