TY - JOUR
T1 - Integration in Hilbert generated Banach spaces
AU - Deville, Robert
AU - Rodri guez, José
N1 - Funding Information:
∗The second-named author was supported by MEC and FEDER MTM2005-08379). Current address: Departamento de Matemática Aplicada, Facultad formática, Universidad de Murcia, 30100 Espinardo (Murcia), Spain. Received May 19, 2008
PY - 2010
Y1 - 2010
N2 - We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.
AB - We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.
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U2 - 10.1007/s11856-010-0047-4
DO - 10.1007/s11856-010-0047-4
M3 - Article
SN - 0021-2172
VL - 177
SP - 285
EP - 306
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -