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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
Summary:
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.
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