Best possible triangle inequalities for statistical metric spaces

Abstract

1. A statistical metric space2 (briefly, an SM space) is a set S and a mapping 5Y from SXS into the set of distribution functions (i.e., real-valued functions of a real variable which are everywhere defined, nondecreasing, left-continuous and have inf 0 and sup 1). The distribution function 5:(p, q) associated with a pair of points (p, q) in S is denoted by Fpq. The functions Fpq are assumed to satisfy: (SM-I) Fpq(x) = 1 for all x>0 if p=q. (SM-II) Fpq(O)=O. (SM4 -III) F,q = Fq. (SM-IV) If Fp,q(x) = 1 and Fqr(y) = 1, then Frr(x+y) = 1. A real-valued function T, whose domain is the set of real number pairs (x, y) such that 0 T(a, b) if c>a, d>b. (T-III) T(a, b) = T(b, a) (commutativity). (T-IV) T [T(a, b), c ] = T [a, T(b, c) ] (associativity). DEFINITION 1. A Menger space (S, iY, T) is an SM space (S, i) and a t-function T such that the triangle inequality, (SM-IVm) F,r(x+y) > T(Fp,q(x), Fqr(y)), holds for all points p, q, r in S and for all numbers x, y>0. DEFINITION 2. The t-function T, is stronger than the t-function T2, and we write T, > T2, if Ti(x, y) > T2(x, y) for O T2(x, y). Correspondingly, T2 is weaker or strictly weaker than T7. For a given SM space there is in general more than one t-function which makes it a Menger space. In particular, if (S, 59, T) is a Menger space and U is weaker than T, then (S, 5Y, U) is also a Menger space