Atom (measure theory)

A measure that has no atoms is called non-atomic or atomless.is the symmetric difference operator.-finite measure, there are countably many atomic classes.This is equivalent to say that there is a countable partition offormed by atoms up to a null set.This space is atomic, with all atoms being the singletons, yet the space is not able to be partitioned into the disjoint union of countably many disjoint atoms,since the countable union of singletons is a countable set, and the uncountability of the real numbers shows that the complement-measure would be infinite, in contradiction to it being a null set.-finite spaces follows from the proof for finite measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null.is the weighted sum of countably many Dirac measures, that is, there is a sequenceof positive real numbers (the weights) such thatA discrete measure is atomic but the inverse implication fails: takeThen there is a single atomic class, the one formed by the co-countable subsets.can't be put as a sum of Dirac measures.If every atom is equivalent to a singleton, thenAny finite measure in a separable metric space provided with the Borel sets satisfies this condition.there exists a measurable subsetA non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a setone can construct a decreasing sequence of measurable setsIt turns out that non-atomic measures actually have a continuum of values.there exists a measurable subsetThis theorem is due to Wacław Sierpiński.[6][7] It is reminiscent of the intermediate value theorem for continuous functions.Sketch of proof of Sierpiński's theorem on non-atomic measures.A slightly stronger statement, which however makes the proof easier, is that ifthat is monotone with respect to inclusion, and a right-inverse toThat is, there exists a one-parameter family of measurable setsThe proof easily follows from Zorn's lemma applied to the set of all monotone partial sections toordered by inclusion of graphs,It's then standard to show that every chain in
mathematicsmeasure theorymeasurable spacemeasureequivalence classsymmetric differencesigma-algebrapower setcardinalitysingletonsLebesgue measurereal linecountablecounting measureuncountability of the real numbersDiscrete measurecontinuumWacław Sierpińskiintermediate value theoremZorn's lemmaAtom (order theory)Dirac delta functionElementary eventAbsolute continuityof measuresLebesgue integrationLp spacesMeasure spaceProbability spacefunctionAlmost everywhereBaire setBorel setequivalence relationBorel spaceCarathéodory's criterionCylindrical σ-algebraCylinder set𝜆-systemEssential rangeinfimum/supremumLocally measurableπ-systemσ-algebraNon-measurable setVitali setNull setSupportTransverse measureUniversally measurablemeasuresAtomicBanachComplexCompleteContentLogarithmicallyConvexDecomposableDiscreteEquivalentFiniteQuasi-InvariantLocally finiteMaximisingMetric outerPerfectPre-measureProbabilityProjection-valuedRandomRegularBorel regularInner regularOuter regularSaturatedSet functionσ-finites-finiteSignedSingularSpectralStrictly positiveVectorCountingGaussianHarmonicHausdorffIntensityLebesgueInfinite-dimensionalProductProjectionsPushforwardSpherical measureTangentTrivialMeasurable functionBochnerStronglyWeaklyin measureof random variablesin distributionin probabilityCylinder set measurecompact setelementprocessvariableProjection-valued measureCarathéodory's extension theoremDominatedMonotoneVitaliJordanMaharam'sEgorov'sFatou's lemmaFubini'sFubini–TonelliHölder's inequalityMinkowski inequalityRadon–NikodymRiesz–Markov–Kakutani representation theoremDisintegration theoremLifting theoryLebesgue's density theoremLebesgue differentiation theoremSard's theoremVitali–Hahn–Saks theoremIsoperimetric inequalityBrunn–Minkowski theoremMilman's reverseMinkowski–Steiner formulaPrékopa–Leindler inequalityVitale's random Brunn–Minkowski inequalityConvex analysisDescriptive set theoryProbability theoryReal analysisSpectral theory