On coincidence of Pettis and McShane integrability

Marián J. Fabián

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 83-106
  • ISSN: 0011-4642

Abstract

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space X , every Pettis integrable function f : [ 0 , 1 ] → X is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [ 0 , 1 ] into X , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [ 0 , 1 ] (mostly) into C ( K ) spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces K , that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from [ 0 , 1 ] into C ( K ) in McShane sense.

How to cite

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Fabián, Marián J.. "On coincidence of Pettis and McShane integrability." Czechoslovak Mathematical Journal 65.1 (2015): 83-106. <http://eudml.org/doc/270052>.

@article{Fabián2015,
abstract = {R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f\colon [0,1]\rightarrow X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C(K)$ spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces $K$, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from $[0,1]$ into $C(K)$ in McShane sense.},
author = {Fabián, Marián J.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Pettis integral; McShane integral; MC-filling family; uniform Eberlein compact space; scalarly negligible function; Lebesgue injection; Hilbert generated space; strong Markuševič basis; adequate inflation; Pettis integral; McShane integral; MC-filling family; uniform Eberlein compact space; scalarly negligible function; Lebesgue injection; Hilbert generated space; strong Markuševič basis; adequate inflation},
language = {eng},
number = {1},
pages = {83-106},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On coincidence of Pettis and McShane integrability},
url = {http://eudml.org/doc/270052},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Fabián, Marián J.
TI - On coincidence of Pettis and McShane integrability
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 83
EP - 106
AB - R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f\colon [0,1]\rightarrow X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C(K)$ spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces $K$, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from $[0,1]$ into $C(K)$ in McShane sense.
LA - eng
KW - Pettis integral; McShane integral; MC-filling family; uniform Eberlein compact space; scalarly negligible function; Lebesgue injection; Hilbert generated space; strong Markuševič basis; adequate inflation; Pettis integral; McShane integral; MC-filling family; uniform Eberlein compact space; scalarly negligible function; Lebesgue injection; Hilbert generated space; strong Markuševič basis; adequate inflation
UR - http://eudml.org/doc/270052
ER -

References

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