A convergence theorem on the dunford integral

Abstract

This article discusses the convergence theorem of the Dunford integrals. We examine the sufficient conditions so that limit of the sequence of integral value whose Dunford integrable is same as limit of functions sequence. We have obtained that to guarantee a function to be Dunford integrable and its limit of functions sequence are same as value of the functions, then a sequence of Dunford integrable function is uniform convergent or weakly convergent, weakly monoton, and its limit exist. Furthermore, its weakly convergent and bounded.

On the McShane-Dunford-Stieltjes Integral and McShane-Pettis-Stieltjes Integral

“…The Pettis integral has two definitions: one for Dunford integrable functions, where the integral coincides with the Dunford integral; the other for weakly measurable functions with Lebesgue integrable images, ensuring the existence of an element satisfying a certain condition for every measurable set [10]. It is also important to note that the convergence theorems of the Dunford integral wherein in a sequence of Dunford integrable function is uniform convergent or weakly convergent, weakly monotone and its limit exists [11]. But Gouju and Schwabik (2002) mentioned that the relation between the Pettis integral and the McShane integral for arbitrary spaces is unknown [5].…”