On measurable multifunctions with applications to random multivalued equations.

*(English)*Zbl 0634.28005Let (\(\Omega\),\({\mathcal A},\mu)\) be a complete, finite measure space, let X be a separable, reflexive Banach space and let \(F: \Omega \times X\to 2^ X\setminus \{\emptyset \}\) be a set-valued function with closed and convex values such that \((i)\quad F(\cdot,x)\) is measurable, \((ii)\quad F(\omega,\cdot)\) is upper semicontinuous (with respect to the weak topology) and \((iii)\quad F(\cdot,\cdot)\) is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that \(F(\cdot,\cdot)\) is \({\mathcal A}\times {\mathcal B}(X)\)- measurable, where \({\mathcal B}(X)\) are the Borel-sets on X.

Now suppose that (\(\Omega\),\({\mathcal A})\) is a measurable space with \({\mathcal A}^ a \)Souslin family and X is a Souslin metric space. Then the measurability of \(\limsup_ nF_ n\) and \(\liminf_ nF_ n\) is proved for a sequence of closed valued, measurable set-valued functions \(F_ n: \Omega \to 2^ X\setminus \{\emptyset \}.\) Other similar results are shown.

The measurability results are used to obtain new implicit function theorems [cf. J. C. Himmelberg, Fundam. Math. 87, 53-72 (1975; Zbl 0296.28003)] and to prove existence of solutions for implicit random differential inclusions and random functional-differential inclusions.

Now suppose that (\(\Omega\),\({\mathcal A})\) is a measurable space with \({\mathcal A}^ a \)Souslin family and X is a Souslin metric space. Then the measurability of \(\limsup_ nF_ n\) and \(\liminf_ nF_ n\) is proved for a sequence of closed valued, measurable set-valued functions \(F_ n: \Omega \to 2^ X\setminus \{\emptyset \}.\) Other similar results are shown.

The measurability results are used to obtain new implicit function theorems [cf. J. C. Himmelberg, Fundam. Math. 87, 53-72 (1975; Zbl 0296.28003)] and to prove existence of solutions for implicit random differential inclusions and random functional-differential inclusions.

Reviewer: W.J.A.Stich

##### MSC:

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

60H25 | Random operators and equations (aspects of stochastic analysis) |

49J27 | Existence theories for problems in abstract spaces |