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<div>In [[measure theory]] (a branch of [[mathematical analysis]]), a property holds '''almost everywhere''' if the set of elements for which the property does not hold is a set of measure zero (Halmos 1974). In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero. When discussing sets of [[real number]]s, the [[Lebesgue measure]] is assumed unless otherwise stated. <br />
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The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent [[French language]] phrase ''presque partout''.<br />
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A set with '''full measure''' is one whose complement is of measure zero. In [[probability theory]], the terms ''[[almost surely]]'', ''almost certain'' and ''almost always'' refer to sets with [[probability]] 1, which are exactly the sets of full measure in a probability space. <br />
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Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for '''almost all''' elements (though the term [[almost all]] also has other meanings).<br />
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== Properties ==<br />
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* If ''f'' : '''R''' → '''R''' is a [[Lebesgue integral|Lebesgue integrable]] function and ''f''(''x'') ≥ 0 almost everywhere, then<br />
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::<math>\int_a^b f(x) \, dx \geq 0</math><br />
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:for all real numbers ''a'' < ''b'' with equality [[iff]] <math>f(x)=0</math> almost everywhere.<br />
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* If ''f'' : [''a'', ''b''] → '''R''' is a [[monotonic function]], then ''f'' is [[derivative|differentiable]] almost everywhere.<br />
* If ''f'' : '''R''' → '''R''' is Lebesgue measurable and<br />
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::<math>\int_a^b |f(x)| \, dx < \infty</math><br />
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:for all real numbers ''a'' < ''b'', then there exists a set ''E'' (depending on ''f'') such that, if ''x'' is in ''E'', the Lebesgue mean<br />
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::<math>\frac{1}{2\epsilon} \int_{x-\epsilon}^{x+\epsilon} f(t)\,dt</math><br />
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:converges to ''f''(''x'') as <math>\epsilon</math> decreases to zero. The set ''E'' is called the Lebesgue set of ''f''. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of ''f'' converges to ''f'' almost everywhere.<br />
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* If ''f''(''x'',''y'') is [[Borel measurable]] on '''R'''<sup>2</sup> then for almost every ''x'', the function ''y''→''f''(''x'',''y'') is Borel measurable.<br />
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* A bounded [[function (mathematics)|function]] ''f'' : [''a'', ''b''] <tt>-></tt> '''R''' is [[Riemann integral|Riemann integrable]] if and only if it is [[continuous function|continuous]] almost everywhere.<br />
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== Definition using ultrafilters ==<br />
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Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an [[ultrafilter]]. An ultrafilter on a set ''X'' is a maximal collection ''F'' of subsets of ''X'' such that:<br />
# If ''U'' ∈ ''F'' and ''U'' ⊆ ''V'' then ''V'' ∈ ''F''<br />
# The intersection of any two sets in ''F'' is in ''F''<br />
# The empty set is not in ''F''<br />
A property ''P'' of points in ''X'' holds almost everywhere, relative to an ultrafilter ''F'', if the set of points for which ''P'' holds is in ''F''. <br />
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For example, one construction of the [[hyperreal number]] system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.<br />
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The definition of ''almost 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.<br />
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==References==<br />
* {{cite book<br />
| last = Billingsley<br />
| first = Patrick<br />
| authorlink = <br />
| year = 1995<br />
| title = Probability and measure<br />
| edition = 3rd edition<br />
| publisher = John Wiley & sons<br />
| location = New York<br />
| isbn = 0-471-00710-2<br />
}}<br />
* {{cite book<br />
| last = Halmos<br />
| first = Paul R.<br />
| authorlink = Paul Halmos<br />
| year = 1974<br />
| title = Measure Theory<br />
| publisher = Springer-Verlag<br />
| location = New York<br />
| isbn = 0-387-90088-8<br />
}}<br />
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[[Category:Measure theory]]<br />
[[Category:Mathematical terminology]]<br />
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[[cs:Skoro všude]]<br />
[[de:Maßtheorie#fast überall]]<br />
[[es:Casi en todas partes]]<br />
[[eo:Preskaŭ ĉie]]<br />
[[ko:거의 모든 점에서]]<br />
[[it:Quasi ovunque]]<br />
[[nl:Bijna overal]]<br />
[[ja:ほとんど (数学)#ほとんど至るところで]]<br />
[[ru:Почти всюду]]<br />
[[fi:Melkein kaikkialla]]<br />
[[sv:Nästan överallt]]<br />
[[uk:Майже скрізь]]<br />
[[zh:幾乎處處]]</div>76.126.195.3