Introduced by Vitali(1907), Hahn(1922) and Saks(1933), the VitaliHahn-Saks theorem plays a vital role in measure theory in proving some weak convergence theorems. In this Toolbox Essay, we will first introduce the Nikodym theorem followed by a self-contained proof. Next, we state the Vitali-Hahn-Saks theorem and show by an example how this theorem can be used in probability theory. The benefit ...

K.Balasubramanian , K. Chandrasekhara rao *2 Department of Mathematics, School of Humanities and Sciences, SASTRA University, Thanjavur 613402, India 1 [email protected] * Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam,India 2 [email protected] Abstract— This paper is devoted to the rate space of Hahn sequence space. It is denoted by . Some properti...

A functional analysis course is taught in almost all mathematics departments with a Ph.D. program. Functional analysis is primarily the study of the algebraic, analytic and geometric structures of infinite-dimensional vector spaces and operators on them. The prototypical example is spaces of functions, regarded as vector spaces, where the operators are differential and integral operators. The s...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...