Inmeasure theory (a branch ofmathematical analysis), a property holdsalmost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept ofmeasure zero, and is analogous to the notion ofalmost surely inprobability theory.

More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero,^[1][2] or equivalently, if the set of elements for which the property holds isconull. In cases where the measure is notcomplete, it is sufficient that the set be contained within a set of measure zero. When discussing sets ofreal numbers, theLebesgue measure is usually assumed unless otherwise stated.

The termalmost everywhere is abbreviateda.e.;^[3] in older literaturep.p. is used, to stand for the equivalentFrench phrasepresque partout.^[4]

A set withfull measure is one whose complement is of measure zero. In probability theory, the termsalmost surely,almost certain andalmost always refer toevents withprobability 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space.

Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds foralmost all elements (though the termalmost all can also have other meanings).

If${\displaystyle (X,\mathrm{\Sigma },\mu )}$ is ameasure space, a property${\displaystyle P}$ is said to hold almost everywhere in${\displaystyle X}$ if there exists a measurable set${\displaystyle N\in \mathrm{\Sigma }}$ with${\displaystyle \mu (N)=0}$, and all${\displaystyle x\in X\setminus N}$ have the property${\displaystyle P}$.^[5] Another common way of expressing the same thing is to say that "almost every point satisfies${\displaystyle P}$", or that "for almost every${\displaystyle x}$,${\displaystyle P(x)}$ holds".

It isnot required that the set${\displaystyle \{x\in X:\mathrm{\neg }P(x)\}}$ has measure zero; it may not be measurable. By the above definition, it is sufficient that${\displaystyle \{x\in X:\mathrm{\neg }P(x)\}}$ be contained in some set${\displaystyle N}$ that is measurable and has measure zero. However, this technicality vanishes when considering acomplete measure space: if${\displaystyle X}$ is complete then${\displaystyle N}$ exists with measure zero if and only if${\displaystyle \{x\in X:\mathrm{\neg }P(x)\}}$ is measurable with measure zero.

• If property${\displaystyle P}$ holds almost everywhere and implies property${\displaystyle Q}$, then property${\displaystyle Q}$ holds almost everywhere. This follows from themonotonicity of measures.
• If${\displaystyle (P_n)}$ is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction${\displaystyle \mathrm{\forall }n{P}_n}$ holds almost everywhere. This follows from thecountable sub-additivity of measures.
• By contrast, if${\displaystyle (P_x{)}_{x\in \mathbf{R}}}$ is an uncountable family of properties, each of which holds almost everywhere, then their conjunction${\displaystyle \mathrm{\forall }x{P}_x}$ does not necessarily hold almost everywhere. For example, if${\displaystyle \mu }$ is Lebesgue measure on${\displaystyle X=\mathbf{R}}$ and${\displaystyle P_x}$ is the property of not being equal to${\displaystyle x}$ (i.e.${\displaystyle P_x(y)}$ is true if and only if${\displaystyle y\ne x}$), then each${\displaystyle P_x}$ holds almost everywhere, but the conjunction${\displaystyle \mathrm{\forall }x{P}_x}$ does not hold anywhere.

As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction.^{[citation needed]} This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".

• Iff :R →R is aLebesgue integrable function and${\displaystyle f(x)\ge 0}$ almost everywhere, then$${\displaystyle {\int }_a^{b}f(x)dx\ge 0}$$ for all real numbers with equalityif and only if${\displaystyle f(x)=0}$ almost everywhere.
• Iff : [a,b] →R is amonotonic function, thenf isdifferentiable almost everywhere.
• Iff :R →R isLebesgue measurable and for all real numbers, then there exists a setE (depending onf) such that, ifx is inE, the Lebesgue mean$${\displaystyle \frac{1}{2\epsilon }{\int }_{x-\epsilon }^{x+\epsilon }f(t)dt}$$ converges tof(x) as${\displaystyle ϵ}$ decreases to zero. The setE is called the Lebesgue set off. Its complement can be proved to have measure zero. In other words, the Lebesgue mean off converges tof almost everywhere.
• A boundedfunctionf : [a,b] →R isRiemann integrable if and only if it iscontinuous almost everywhere.
• As a curiosity, the decimal expansion of almost every real number in the interval [0, 1] contains the complete text ofShakespeare's plays, encoded inASCII; similarly for every other finite digit sequence, seeNormal number.

Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of anultrafilter. An ultrafilter on a setX is a maximal collectionF of subsets ofX such that:

1. IfU ∈F andU ⊆V thenV ∈F
2. The intersection of any two sets inF is inF
3. The empty set is not inF

A propertyP of points inX holds almost everywhere, relative to an ultrafilterF, if the set of points for whichP holds is inF.

For example, one construction of thehyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.

The definition ofalmost everywhere in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.

• Dirichlet's function, a function that is equal to 0 almost everywhere.
• Cantor function

• Billingsley, Patrick (1995).Probability and measure (3rd ed.). New York: John Wiley & Sons.ISBN 0-471-00710-2.

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• Measure theory

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