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Session 36. Topology in Functional Analysis
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On a Gulisashvili question on scalarly measurable functions |
Anatolij Plichko, Cracow University of Technology, Poland
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Let \(X\) be a Banach space, \(X^*\) be its dual and \((\Omega,
\Sigma)\) be a measurable space. A function \(\varphi :\Omega\to X\)
is scalarly measurable if \(f\circ \varphi\) is measurable for
every \(f\in X^*\). The \(\varphi\) is totally scalarly
measurable if the set
\[F=\{f\in X^*:f\circ \varphi\;\;\mbox{is measurable}\}\]
is total, i.e. for every \(x\in X\), \( \, x\ne 0\), there exists \(f\in
F\) such that \(f(x)\ne 0\). A Banach space \(X\) satisfies the
\textit{property} \(\mathcal{D}\) if for every measurable space
\((\Omega, \Sigma)\) every totally scalarly measurable function
\(\varphi :\Omega\to X\) is, in fact, scalarly measurable.\medskip
Property \(\mathcal{D}\) has been introduced by A. Gulisashvili
[1] in connection with the Pettis integral in interpolation
spaces. Gulisashvili has proved that the weak* angelicity of \(X^*\)
implies the property \(\mathcal{D}\) of \(X\) and has risen the
problem of the reverse implication. We answer the Gulisashvili
problem in negative.
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References- A. Gulisashvili, Estimates for the Pettis integral in
interpolation spaces and inversion of the embedding theorems ,
Dokl. Acad. Nauk SSSR 263, 1982, 793--798 (Russian). English
transl.: Sovet. Math. Dokl. 25, 1982, 428-432.
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