On coincidence of Pettis and McShane integrability

Czechoslovak Mathematical Journal

Marián Fabian¹ 

[1] Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

关键词: Pettis integral;    McShane integral;    MC-filling family;    uniform Eberlein compact space;    scalarly negligible function;    Lebesgue injection;    Hilbert generated space;    strong Markuševič basis;    adequate inflation;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

R. Deville and J. Rodriguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f [0,1]\rightarrow X$ 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.

【 授权许可】

Unknown