The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (, , P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX _pq with the interpretation thatX _pq () is the distance betweenp andq at the instant . The probability of the eventJ _pqr = { :X _pr ()X _pq ()+X _qr ()} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, Statistical Metrics, Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, Statistical Metric Spaces, Pacific J. Math.10 (1960), 313–334). It turns out that for > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X < Y + Z) < . This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J _pqr ) = 1.