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Línia 1: 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 NULL SET , that 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 NUMBERS , the LEBESGUE MEASURE is assumed unless otherwise stated. .. ~~{{Petició de traducció\|en\|Almost everywhere\|Usuari:Amical-bot/Matemàtiques/en\|--[[Usuari:Gomà\|Gomà]] ([[Usuari Discussió:Gomà\|disc.]]) 10:01, 16 gen 2010 (CET)}}~~ .. 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 )) . .. .. 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. .. .. 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). .. .. (==) Properties (==) .. .. * If (( f )) : ((((( R ))))) → ((((( R ))))) is a LEBESGUE INTEGRABLE function and (( f )) (((( x )))) ≥ 0 almost everywhere , then .. .. :: <math>\int_a^b f(x) \ , dx \geq 0</math> .. .. : for all real numbers (( a )) < (( b )) . .. .. * If (( f )) : [(( a )) , (( b )) ] → ((((( R ))))) is a MONOTONIC function , then (( f )) is DIFFERENTIABLE almost everywhere. .. * If (( f )) : ((((( R ))))) → ((((( R ))))) is Lebesgue measurable and .. .. :: <math>\int_a^b .3. f(x) .3. \ , dx < \infty</math> .. .. : 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 .. .. :: <math>\frac{1}{2\epsilon} \int_{x-\epsilon}^{x+\epsilon} f(t)\ ,dt</math> .. .. : converges to (( f )) (((( x )))) as <math>\epsilon</math> decreases to zero. The set (( E )) is called the Lebesgue set of (( f )) . Its compliment can be proved to have measure zero. In other words , the Lebesgue mean of (( f )) converges to (( f )) almost everywhere. .. .. * 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. .. .. * A bounded FUNCTION (( f )) : [(( a )) , (( b )) ] <tt>-></tt> ((((( R ))))) is RIEMANN INTEGRABLE if and only if it is CONTINUOUS almost everywhere. .. .. (==) Definition using ultrafilters (==) .. .. 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: .. # If (( U )) &isin ; (( F )) and (( U )) &sube ; (( V )) then (( V )) &isin ; (( F )) .. # The intersection of any two sets in (( F )) is in (( F )) .. # The empty set is not in (( F )) .. A property of points in (( X )) holds almost everywhere , relative to an ultrafilter (( F )) , if the set of points for which (( X )) holds is in (( F )) . .. .. 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. .. .. 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. .. .. (==) References (==) .. * {{ cite book .. .3. last = Billingsley .. .3. first = Patrick .. .3. authorlink = .. .3. year = 1995 .. .3. title = Probability and measure .. .3. edition = 3rd edition .. .3. publisher = John Wiley & sons .. .3. location = New York .. .3. isbn = 0-471-00710-2. .. }} .. .. * {{ cite book .. .3. last = Halmos .. .3. first = Paul R. .. .3. authorlink = Paul Halmos .. .3. year = 1974 .. .3. title = Measure Theory .. .3. publisher = Springer-Verlag .. .3. location = New York .. .3. isbn = 0-387-90088-8 .. }} .. .. [[Category:Measure theory]] .. [[Category:Mathematical terminology]] .. .. [[de:Fast überall]] .. [[eo:Preskaŭ ĉie]] .. [[fr:Presque partout]] .. [[it:Quasi ovunque]] .. [[ja:ほとんど至る所]] .. [[nl:Bijna overal]] .. [[pl:Zbiór miary zero]] .. [[ru:Почти всюду]] .. [[fi:Melkein kaikkialla]] .. [[sv:Nästan överallt]] .. [[zh:幾乎處處]] .. paraulesenllacos .. .. MEASURE THEORY .. .. MATHEMATICAL ANALYSIS .. .. NULL SET .. .. MEASURE ZERO .. .. REAL NUMBERS .. .. LEBESGUE MEASURE .. .. FRENCH LANGUAGE .. .. PROBABILITY THEORY .. .. ALMOST SURELY .. .. PROBABILITY .. .. ALMOST ALL .. .. LEBESGUE INTEGRABLE .. .. MONOTONIC .. .. DIFFERENTIABLE .. .. BOREL MEASURABLE .. .. FUNCTION .. .. RIEMANN INTEGRABLE .. .. CONTINUOUS .. .. ULTRAFILTER .. .. HYPERREAL NUMBER ..

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