On the \(T\)-integration of Karták and Mařík.

Abstract

General notions of integration have been introduced by Saks \cite[p. 254]{S3}, Kart\'ak \cite[p. 482]{K14}, Kubota \cite[p. 389]{K8} and Sarkhel \cite[p. 299]{S19}. Kart\'ak's \(T\)-integration was further studied by Kart\'ak and Ma\v{r}\`{\i} in \cite{K15}, and by Kubota in \cite{K17}. \par In this paper, starting from Kartak and Ma\v{r}\`{\i}'s definition, we introduce another general integration (see Definition \ref{D3}), that allows a very general theorem of dominated convergence (see Theorem \ref{T2}). Then we present a general definition for primitives, and this definition contains many of the known nonabsolutely convergent integrals: the Denjoy\(^*\)-integral, the \(\alpha\)-Ridder integral, the wide Denjoy integral, the \(\beta\)-Ridder integral, the Foran integral, the AF integral, the Gordon integral. Using this integration and Theorem \ref{T2}, we obtain a generalization of a result on differential equations, of Bullen and Vyborny \cite{B17}. \par We further give a Banach-Steinhaus type theorem, a categoricity theorem, Riesz type theorems (as a particular case we obtain the Alexiewicz Theorem \cite{A2}), and study the weak convergence for the \(T\)-integration.