Pettis integral

In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality.The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure.is a normed space or (more generally) is a Hausdorff locally convex TVS.Evaluation of a functional may be written as a duality pairing:is called weakly measurable if for allA weakly measurable mapCommon notations for the Pettis integralTo understand the motivation behind the definition of "weakly integrable", consider the special case whereIn this case, every linear functionalis just scalar multiplication by a constant), the conditionis identified as a vector subspace of the double dualis a semi-reflexive space if and only if this map is surjective.An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: Iffor real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminormsThe right-hand side is the lower Lebesgue integral of aTaking a lower Lebesgue integral is necessary because the integrandThis follows from the Hahn-Banach theorem because for every vectorAn important property is that the Pettis integral with respect to a finite measure is contained in the closure of the convex hull of the values scaled by the measure of the integration domain:This is a consequence of the Hahn-Banach theorem and generalizes the mean value theorem for integrals of real-valued functions: If, then closed convex sets are simply intervals and fora Borel measure that assigns finite values to compact subsets,is quasi-complete (that is, every bounded Cauchy net converges) and ifis weakly measurable and there exists a compact, convexbe a sequence of Pettis-integrable random variables, and writedenote the sample average.Suppose that the partial sumsconverge absolutely in the topology ofin the sense that all rearrangements of the sum converge to a single vectorThe weak law of large numbers implies that[citation needed] To get strong convergence, more assumptions are necessary.

mathematicsIsrael M. GelfandBilly James PettisLebesgue integralmeasure spacedualityLebesgue measureBochner integraltopological vector spacenormed spacelocally convexduality pairingmeasurable mapLebesgue integrablesemi-reflexive spacesurjectiveHahn-Banach theoremconvex hullBorel- σ {\displaystyle \sigma } -algebraBorel measurequasi-completeCauchy netweak topologyBochner measurable functionBochner spaceVector measureWeakly measurable functionProceedings of the National Academy of Sciences of the United States of AmericaIsrael M. Gel'fandMichel TalagrandEncyclopedia of MathematicsEMS PressIntegralsRiemann integralBurkill integralDaniell integralDarboux integralHenstock–Kurzweil integralHaar integralHellinger integralKhinchin integralKolmogorov integralLebesgue–Stieltjes integralPfeffer integralRiemann–Stieltjes integralRegulated integralSubstitutionTrigonometricWeierstrassBy partsPartial fractionsEuler's formulaInverse functionsChanging orderReduction formulasParametric derivativesContour integrationLaplace's methodNumerical integrationSimpson's ruleTrapezoidal ruleRisch algorithmImproper integralsGaussian integralDirichlet integralcompleteincompleteBose–Einstein integralFrullani integralCommon integrals in quantum field theoryStochastic integralsItô integralRusso–Vallois integralStratonovich integralSkorokhod integralBasel problemEuler–Maclaurin formulaGabriel's hornIntegration BeeProof that 22/7 exceeds πWashersShellsAnalysistopological vector spacesAbstract Wiener spaceClassical Wiener spaceConvex seriesCylinder set measureInfinite-dimensional vector functionMatrix calculusVector calculusDerivativesDifferentiable vector–valued functions from Euclidean spaceDifferentiation in Fréchet spacesFréchet derivativeFunctional derivativeGateaux derivativeDirectionalGeneralizations of the derivativeHadamard derivativeHolomorphicQuasi-derivativeBesov measureCanonical GaussianClassical Wiener measureMeasureset functionsProjection-valuedVectorBochnerWeaklyStronglymeasurable functionRadonifying functionDirect integralDunfordRegulatedPaley–WienerCameron–Martin theoremInverse function theoremNash–Moser theoremFeldman–Hájek theoremNo infinite-dimensional Lebesgue measureSazonov's theoremStructure theorem for Gaussian measuresCrinkled arcCovariance operatorFunctional calculusBorel functional calculusContinuous functional calculusHolomorphic functional calculusBanach manifoldbundleConvenient vector spaceChoquet theoryFréchet manifoldHilbert manifoldFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setBanach–Mazur distanceTomita–Takesaki theory