An Extension of the Birkhoff Integrability for Multifunctions

Abstract

Abstract.A comparison between a set-valued Gould type and simple Birkhoff integrals of bf (X)-valued multifunctions with respect to a nonnegative set functionis given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

“…Accordingly with the comparison between Gould and Birkoff integrals given in [28] we have that Birkhoff, Gould, RL integrals of the bounded single valued functions agree in the countably additive case, see [28] (Theorem 3.10), while in [43] (Remark 5.5) an analogous comparison is given with the Choquet integral.…”

“…In the literature several methods of integration for functions and multifunctions have been studied extending the Riemann and Lebesgue integrals. In this framework a generalization of Riemann sums was given in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] while another generalization is due to Kadets and Tseytlin [38], who introduced the absolute Riemann-Lebesgue |RL| and unconditional Riemann-Lebesgue RL integrability, for Banach valued functions with respect to countably additive measures. They proved that in finite measure space, the Bochner integrability implies |RL| integrability which is stronger than RL integrability that implies Pettis integrability.…”

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

“…Remark 2.4. If µ is σ-finite then the Bi 1 integrability is correspondent to the classic Birkhoff integrability for Banach space-valued mappings (see also [5,Theorem 3.18]). Moreover, since the Birkhoff integral is stronger than the Pettis integral, it is clear that, as soon as F is first type Birkhoff integrable with respect to m, the mapping M := A → (Bi 1 ) A F dm is a countably additive measure.…”

Some new results on integration for multifunction

“…This means that lim sup This property in [6] is called simple-Birkhoff integrability, and is proved in [6,Theorem 3.18] to be equivalent to Birkhoff integrability of f .…”

“…In the last decades however gauge (non-absolute) integrals have been also considered [2, 3, 7, 8, 10-12, 17, 19, 21, 23, 24, 28, 38] after the pioneering studies of G. Birkhoff, J. Kurzweil, R. Henstock and E. J. McShane. The Birkhoff integral was introduced in [1] and recently investigated in [6,13,14,30,38]. B. Cascales, V. M. Kadets, M. Potyrala, J. Rodriguez, and other authors considered the unconditional Riemann-Lebesgue multivalued integral.…”