6533b7d8fe1ef96bd126b86d

RESEARCH PRODUCT

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

José Rodríguez

subject

Discrete mathematicsPettis integralPure mathematicsMcShane integralIntegrable systemApplied MathematicsBanach spaceProjectional resolution of the identitySeparable spaceAbsolutely summing operatorScalarly null functionWeakly Lindelöf determined Banach spacePettis integralEquivalence (measure theory)Continuum hypothesisAnalysisMathematicsProperty (M)

description

Abstract We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelof determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.

yearjournalcountryeditionlanguage
2009-02-01Journal of Mathematical Analysis and Applications
10.1016/j.jmaa.2007.10.076http://dx.doi.org/10.1016/j.jmaa.2007.10.076