Almost All

analysis set-theory

Definition

Almost All

A property is said to hold for almost all elements of a set if the number of exceptions is finite.

Formally, for a set $X$, a property $P$ holds for almost all $x\in X$ if the set $\{x\in X\mid \neg P(x)\}$ is a finite set.

Contextual Usage

Analysis

In the context of sequences, a property holds for almost all $n\in \mathbb{N}$ if there exists some index $N$ such that the property holds for all $n>N$. This is a fundamental concept in the definition of limits and convergence.

Measure Theory

Note that in measure theory, the term “almost everywhere” is used differently. A property holds almost everywhere if the set of exceptions has measure zero, which can include infinitely many (even uncountably many) elements.