Let (T,r,p) be a finite measure space, X be a Banach space, P be a metric space and let L,(y,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T, T, p). We show that if $I: T x P-2x is a set-valued function such that for each fixed p E P, 4(. , p) has a measurable graph and for each fixed TV T, 4(t;) is either upper or lower semicontinuous then the Aumann i...

On the background of a brief survey panorama results on topic in title, one new theorem is presented concerning positive topological entropy (i.e. chaos) for impulsive differential equations Cartesian product compact intervals, which positively invariant under composition associated Poincaré translation operator with multivalued upper semicontinuous mapping.

We show that every rigid motion invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, the length, the area, and a suitable curvature integral of the convex disc. 1991 AMS subject classification: 52A10, 53A04

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of t...