Vector-valued McShane and strong McShane integrals with the Alexiewicz norm and its Riesz representation theorems

Ponente(s): Tomás Pérez Becerra, Juan Alberto Escamilla Reyna, Hemanta Kalita

In this talk, we discuss McShane and strong McShane integration theory in the vector-valued functions endowed with the Alexiewicz norm, their relations, and properties as the fundamental theorem of calculus. We extend this theory to some Stieltjes-type integrals via a bounded bilinear operator, such as McShane-Stieltjes, strong McShane-Stieltjes, and Riemann-McShane-Stieltjes and give some relations with the non-Stieltjes integrals via the integration by parts. Finally, we show some versions of the Riesz representation Theorem for each space of functions that characterizes the respective dual spaces of the McShane and strong McShane integrable functions.