Some new results on integration for multifunction

Abstract

It has been proven in [20] and [18] that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable (Theorem 3.4).keywords Multifunction, set-valued Pettis integral, set-valued variationally Henstock and Birkhoff integrals, selection M.S.

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

“…In particular, comparison of different generalizations of Lebesgue integral is, in our opinion, one of the milestones of the modern theory of integration. Inspired by [6,7,10,12,13,19,24,39], we continue in this paper the study on this subject and we examine relationship among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable.…”

Decompositions of Weakly Compact Valued Integrable Multifunctions

“…Of course, if 0 ∈ Γ(t) for almost every t ∈ [0, 1], then Γ is positive. As regards other definitions of measurability and integrability that are treated here and are not explained and the known relations among them, we refer to [3,[15][16][17][18][19][20]26,36,38,[40][41][42], in order not to burden the presentation.…”

“…Remark 1. Observe that, using the Rådström embedding: i : cwk(X) → l ∞ (B X * ) (see for example [43] or [19])…”

A Vector Girsanov Result and its Applications to Conditional Measures via the Birkhoff Integrability

“…If we consider, for example, a Brownian motion (w t ) t≥0 , we know that this process is a martingale on this space with respect to this filtration F. However, if we condition it on an expiration time T > 0 fixed, the distribution of this process changes and in general, it doesn't preserve some of its properties, such as the martingale property (see for example [15,Theorem 5.4]). Let (7) N := P (·|w T ) : A → L 1 (Ω) .…”