The integral theory of Ioffe's fans (Q1075456)

This paper is concerned with fans [\textit{A. D. Ioffe}, Trans. Am. Math. Soc. 266, 1-56 (1981)] depending measurably on a parameter \(\omega\) (one micht call them ''random fans'') - that is, with closed convex valued multifunctions F defined on \(\Omega\) \(\times X\) (where (\(\Omega\),\(\Sigma\),\(\mu)\) is a complete \(\sigma\)-finite measure space) such that \(x\to F(\omega,x)\) is a fan for each \(\omega\in \Omega\), while \(\omega\) \(\to F(\omega,x)\) is a measurable multifunction for each \(x\in X\). The author studies the rules for computing the Aumann integrals \(\Phi (x)=\int_{\Omega}F(\omega,x)d\mu (\omega).\) He gives mild conditions under which \(\Phi\) turns out to be a fan, whose adjoint \(\Phi^*\) equals the Aumann integral of the adjoint \(F^*\). Next he derives formulas for the Aumann integrals of the ''upper bound'' of two fans, the conditional expectation of a fan, and the adjoints of these quantities. In the final section he treats integrals of the composed fans \(G\circ F\) and \(F\circ G\), where G is an \(\omega\)-independent fan of the special sort generated by a compact set of linear operators. An inclusion extending Jensen's inequality to the multivalued case is also given.