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