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.

We shall denote the set of all regularly varying functions by R. We shall also denote by L the set of slowly slowly varying (or slowly oscillating) functions, namely those functions in R for which the index ρ = 0. It is easy to show that if h ∈ R, then there exists L ∈ L such that h(x) = xL(x), with ρ being the index of h. Slowly varying functions arise naturally in many branches of number theo...

Let L(x) be a slowly varying function at both zero and infinity. The existence of the non-commutative neutrix convolution product of the distributions x+L(x) and x μ − is proved, where λ, μ are real numbers such that λ, μ / ∈ −N and λ+μ / ∈ −Z . Some other products of distributions are obtained. AMS Mathematics Subject Classification (2000): 46F10