in this paper, we introduce the  probabilistic normed groups. among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. we also study midconvex functions defined  on probabilistic normed groups and  give  some results about locally boundedness of such  functions.

A real-valued function f of a real variable is said to be (p-slowly varying ((p-s .v.) if limn_ . rp (x) [ f (x + a) f (x)] = 0 for each a. It is said to be uniformly 9-slowly varying (u . (P-s .v .) if limn-. . sup, e r rp(x) ; f (x-a) f (x)I =0 for every bounded interval I. It is supposed throughout that rp is positive and increasing . It is proved that if w increases rapidly enough, then eve...

This paper is a sequel to both Ash, Erd1⁄2os and Rubel [AER], on very slowly varying functions, and [BOst1], on foundations of regular variation. We show that generalizations of the Ash-Erd1⁄2os-Rubel approach –imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property – lead naturally to the main result of regular variation, th...