Geodesic
On coincidence of Pettis and McShane integrability
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 83-106
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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.
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.
DOI : 10.1007/s10587-015-0161-x
Classification : 46B26, 46G10
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 
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     title = {On coincidence of {Pettis} and {McShane} integrability},
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Fabian, Marián. On coincidence of Pettis and McShane integrability. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 83-106. doi: 10.1007/s10587-015-0161-x

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