Pointwise limits of Birkhoff integrable functions

Abstract

Abstract. We study the Birkhoff integrability of pointwise limits of sequences of Birkhoff integrable Banach space-valued functions, as well as the convergence of the corresponding integrals. Both norm and weak convergence are considered. We discuss the roles that equi-Birkhoff integrability and the Bourgain property play in these problems. Incidentally, a convergence theorem for the Pettis integral with respect to the norm topology is presented.

Integration in Hilbert generated Banach spaces

“…This notion of integrability plays an interesting role in vector integration; see, for instance, [3,14,23,25,26,29,28]. For a function f : Ω → X we always have Bochner integrable =⇒ Birkhoff integrable =⇒ Pettis integrable and the respective integrals coincide.…”

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

“…Here we generalize the Girsanov Theorem to the case of vector measure spaces following the idea formulated in [31] for the real Brownian motion and using the Birkhoff vector integral studied in [12,14,16,19,20,[22][23][24][25]30,[34][35][36][37][38], in [17] for non additive-measures and in [3-9, 13, 18, 21] for the multivalued integration. Other results on the Brownian motion subject are given also in [10,26,29].…”

Convergence theorems for Pettis integral of functions taking values in locally convex spaces

“…Theorem 2.2 has been proved by another approach in [7]. Theorem 2.3 is the analogue of Theorem 2.8 in [9]. For more information on convergence theorems for Pettis integral, see [6], [7], [8], [4] and [2].…”