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Birkhoff integral for multi-valued functions. (English) Zbl 1066.46037
In this paper, \(X, X^{*}\) denotes a Banach space and its topological dual and \(B_{X}, B_{X^{*}} \) the closed unit balls of \(X, X^{*}\); \( (\Omega , \Sigma, \mu)\) is a complete finite measure space, \(cwk(X)\) is the family of all non-empty, weakly compact, convex subsets of \(X\) with Hausdorff distance \(h\), \(ck(X)\) is the family of all non-empty, 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 \( j: cwk(X) \to \ell_{\infty}(B_{X^{*}}) \), with \[ j(A)(x^{*})= \sup \{ x^{*} (x): x \in A \} , \] which is additive, positively homogeneous, distance preserving and with \( j(cwk(X))\) closed in \(\ell_{\infty}(B_{X^{*}}) \). For a Banach space \(Y\), an \( f: \Omega \to Y \) is said to be summable with respect to a countable partition \(\Gamma = \{ A_{n} \} \) of \(\Omega \) if \(f_{| A_{n}} \) is bounded whenever \(\mu( A_{n}) > 0 \) and the set \(J(f, \Gamma)= \{ \sum_{n} \mu ( A_{n}) f(t_{n}): t_{n} \in A_{n} \}\) consists of unconditionally convergent series. If for every \( \varepsilon >0\) there is a countable partition \( \Gamma\) of \(\Omega \) in \( \Sigma \) for which \(f\) is summable and \( \| . \| - \text{diam} (J(f, \Gamma)) \leq \varepsilon \), then the Birkhoff integral is \[ (B) \int_{\Omega} f \,d \mu = \bigcap \{ \overline{co(J(f, \Gamma))} : f \text{ summable with respect to } \Gamma \}. \] Birkhoff integrability lies between Pettis and Bochner integrability. A multi-valued \( F : \Omega \to cwk(X) \) is called Birkhoff integrable if \( j \circ F : \Omega \to \ell_{\infty}(B_{X^{*}}) \) is Birkhoff integrable. First, the authors give some equivalent characterizations of Birkhoff integrability and then prove that this definition is independent of any choice of embedding of \( cwk(X)\) into a Banach space. Then some relations between Pettis, Debreu, and Birkhoff integrability are established. The main results are:
(i) If \( F : \Omega \to cwk(X) \) is Debreu integrable, then \(F\) is Birkhoff integrable.
(ii) If \(X\) is separable and \( F : \Omega \to cwk(X) \) is Birkhoff integrable, then \(F\) is Pettis integrable.
(iii) Suppose that \(X\) is separable and \( F : \Omega \to cwk(X) \) is such that \(F(\Omega)\) is \(h\)-separable. Then \(F\) is Birkhoff integrable if and only if it is Pettis integrable.
(iv) If \(X\) is finite-dimensional, then \( F : \Omega \to ck(X) \) is Debreu integrable if and only if it is Birkhoff integrable.
(v) Suppose that \(X^{*}\) has the Radon-Nikodym property. Then \(X\) is finite-dimensional if and only if every bounded Birkhoff integrable \( F : [0, 1] \to cwk(X) \) is Debreu integrable.
If \(X\) is infinite-dimensional and \(X^{*}\) is separable, the authors give two examples: first an \( F : [0, 1] \to cwk(X) \) which is bounded, Birkhoff integrable but not Debreu integrable; then an \( F : [0, 1] \to cwk(X) \) which is bounded, Pettis integrable, but not Birkhoff integrable.

MSC:
46G10 Vector-valued measures and integration
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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