Definition:Almost All
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Definition
Measure Space
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.
Set Theory
Countable
Let $S$ be a countably infinite set.
Let $P: S \to \set {\text {true}, \text {false} }$ be a property of $S$ such that:
- $\set {s \in S: \neg \map P s}$
is finite.
Then $P$ holds for almost all of the elements of $S$.
Uncountable
Let $S$ be an uncountably infinite set.
Let $P: S \to \set {\text {true}, \text {false} }$ be a property of $S$ such that:
- $\set {s \in S: \neg \map P s}$
is countable (either finite or countably infinite).
Then $P$ holds for almost all of the elements of $S$.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): almost all or almost everywhere
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): almost all (almost everywhere)
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): almost all (almost everywhere)