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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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