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The Pettis integral for multi-valued functions via single-valued ones. (English) Zbl 1119.28009

\(X\) is a Banach space over reals, \(X^{*}\) its topological dual and \(B_{X}, B_{X^{*}} \) the closed unit balls of \(X, X^{*}\); \( (\Omega , \Sigma, \mu)\) is a complete probability space, cwk\((X)\) is the family of all non-empty, weakly compact, convex subsets of \(X\) with Hausdorff distance \(h\), and \( \ell_{\infty}(B_{X^{*}}) \) the Banach space of all bounded real-valued functions on \(B_{X^{*}}\). It is well-known that there is a mapping \( j: \text{cwk}(X) \to \ell_{\infty}(B_{X^{*}}) \), with \( j(A)(x^{*})= \sup \{ x^{*} (x): x \in A \} \), which is additve, positively homogeneous, distance preserving and with \( j(\text{cwk}(X))\) closed in \(\ell_{\infty}(B_{X^{*}}) \). For a Banach space \(Y\) and a function \(f: \Omega \to Y\), \(Z_{f}= \{ y^{*} \circ f: y^{*} \in B_{Y^{*}} \}\). For each \(x^{*} \in X^{*}\) and a \( C \in \text{cwk}(X)\), \(\delta^{*} (x^{*}, C)= \sup x^{*} (C)\). For a multi-function \(F: \Omega \to \text{cwk}(X)\), \(\delta^{*} (x^{*}, F): \Omega \to R\), \(\delta^{*} (x^{*}, F)( \omega)= \delta^{*} (x^{*}, F(\omega))\) and \(W_{F}= \{ \delta^{*} (x^{*}, F): x^{*} \in B_{X^{*}} \}\).
An \(F: \Omega \to \text{cwk}(X)\) is said to be Pettis-integrable if (i) for each \(x^{*} \in X^{*}\), \(\delta^{*} (x^{*}, F)\) is \(\mu\)-integrable, and (ii) for each \(A \in \Omega\), there is a \(\int_{A} F d \mu \in \text{cwk}(X)\) such that \(\delta^{*} (x^{*}, \int_{A} F d \mu ) = \int_{A} \delta^{*} (x^{*}, F) d \mu )\). The authors give some characterizations of Pettis-integrability of multi-functions \(F: \Omega \to \text{cwk}(X)\). Some major results are:
I. The following are equivalent:
(i) For any Pettis-integrable multi-function \(F: \Omega \to \text{cwk}(X)\), the composition \(j \circ F\) is Pettis integrable;
(ii) \((\text{cwk}(X), h)\), \(h\) being the Hausdorff metric, is separable;
(iii) \(X\) has Schur property.
II. Let \( K \in \text{cwk}(X)\). Then the following are equivalent:
(i) \(K\) is norm compact;
(ii) the family \(\{ C \in \text{cwk}(X): C \subset K \} \) is \(h\)-separable;
(iii) for any Pettis-integrable multi-function \(F: \Omega \to \text{cwk}(X)\), with \(F(\omega) \subset K, \forall \omega\), \( j \circ F\) is Pettis-integrable.
III. Let \(X\) be a seperable Banach space, and \(F: \Omega \to \text{cwk}(X)\) a Pettis-integrable multi-function such that every countable subset of \(W_{F}\) is stable. Then \( j \circ F\) is Pettis-integrable and \(Z_{f}\) is stable and uniformly integrable.

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
46G10 Vector-valued measures and integration
47N50 Applications of operator theory in the physical sciences
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