probabilistic metric space

Recall that a metric space is a set X equipped with a distance function d:X×X[0,), such that

  1. 1.

    d(a,b)=0 iff a=b,

  2. 2.

    d(a,b)=d(b,a), and

  3. 3.


In some real life situations, distance between two points may not be definite. When this happens, the distance function d may be replaced by a more general function F which takes any pair of points (a,b) to a distribution functionMathworldPlanetmath F(a,b). Before precisely describing how this works, we first look at the properties of these F(a,b) should have, and how one translates the triangle inequalityMathworldMathworldPlanetmath in this more general setting.

distance distribution functions. Since we are dealing with the distance between a and b, the distribution function F(a,b) must have the property that F(a,b)(0)=0. Any distribution function F such that F(0)=0 is called a distance distribution function. The set of all distance distribution functions is denoted by Δ+. For example, for any r0, the step functionsPlanetmathPlanetmath defined by

er(x) = {0whenxr,1otherwise

are distance distribution functions.

In additionPlanetmathPlanetmath to F(a,b) being a distance distribution function, we need that F(a,b)=e0 iff a=b and F(a,b)=F(b,a). These two conditions correspond to the first two conditions on d.

triangle functions. Finally, we need to generalize the binary operation + so it works on the set of distance distribution functions. Clearly, ordinary addition won’t work as the sum of two distribution functions is no longer a distribution function. Šerstnev developed what is called a triangle function that will do the trick.

First, partial order Δ+ by FG iff F(x)G(x) for all x. It is not hard to see that exey iff yx and that e0 is the top element of Δ+. From the poset Δ+, call a binary operator τ on Δ+ a triangle function if τ turns Δ+ into a partially ordered ( commutative monoid with e0 serving as the identity elementMathworldPlanetmath. Spelling this out, for any F,G,HΔ+, we have

  • FτG=GτF,

  • (FτG)τH=Fτ(GτH),

  • Fτe0=e0τF=F, and

  • if GH, then FτGFτH,

where FτG means τ(F,G). For example, FτG=FG, FτG=min(F,G) are two triangle functions. In fact, since FτGFτe0=F and FτGG similarly, we have FτGmin(F,G) for any triangle function τ.

With this, we are ready for our main definition:

Definition. A probabilistic metric space is a (non-empty) set X, equipped with a function F:X×XΔ+, where Δ+ is the set of distance distribution functions on which a triangle function τ is defined, such that

  1. 1.

    F(a,b)=e0 iff a=b, where F(a,b):=F(a,b),

  2. 2.

    F(a,b)=F(b,a), and

  3. 3.


Given a metric space (X,d), if we can find a triangle function τ such that exτey=ex+y, then (X,F) with F(a,b):=ed(a,b) is a probabilistic metric space.


  • 1 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier Science Publishing Company, (1983).
  • 2 A. N. Šerstnev, Random normed spaces: problems of completeness, Kazan. Gos. Univ. Učen. Zap. 122, 3-20, (1962).
Title probabilistic metric space
Canonical name ProbabilisticMetricSpace
Date of creation 2013-03-22 16:49:38
Last modified on 2013-03-22 16:49:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 54E70
Defines distance distribution function
Defines triangle function