almosteverywhere

Almost everywhere, abbreviated a.e., is a term used in measure theory and related areas of mathematics to describe a property that holds for all points of a given space except on a subset of measure zero.

In a measure space (X, F, μ), a function f: X → R is said to equal another function g almost everywhere if μ({x ∈ X : f(x) ≠ g(x)}) = 0. A property P(x) holds almost everywhere if the set {x ∈ X : P(x) is false} has measure zero. The standard example uses Lebesgue measure on the real numbers.

Deriving from this is the probabilistic notion almost surely: a property holds almost surely if the complement set has probability zero. The two notions align when μ is a probability measure, so almost everywhere and almost surely are often used interchangeably in practice.

Examples: The function f(x) = 0 for all x ≠ 0 in [−1,1] and f(0) = 1 equals zero almost everywhere, since the exception set {0} has measure zero. The characteristic function of the rational numbers in [0,1] is 1 on rationals and 0 on irrationals; it equals zero almost everywhere because the rationals form a measure-zero set.

Consequences and uses: If f = g almost everywhere, then their integrals with respect to μ are equal. Almost everywhere convergence describes a form of pointwise convergence that holds except on a measure-zero set; many theorems in integration and analysis (such as dominated convergence and Fatou’s lemma) relate almost everywhere convergence to convergence of integrals. The notion depends on the underlying measure and space, and it does not guarantee equality at every point or regularity properties like continuity.

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