probabilistic metric space
Recall that a metric space is a set $X$ equipped with a distance function $d:X\times X\to [0,\mathrm{\infty})$, such that

1.
$d(a,b)=0$ iff $a=b$,

2.
$d(a,b)=d(b,a)$, and

3.
$d(a,c)\le d(a,b)+d(b,c)$.
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 function^{} ${F}_{(a,b)}$. 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 inequality^{} 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 ${\mathrm{\Delta}}^{+}$. For example, for any $r\ge 0$, the step functions^{} defined by
${e}_{r}(x)$  $=$  $\{\begin{array}{cc}0\hfill & \text{when}x\le r,\hfill \\ 1\hfill & \text{otherwise}\hfill \end{array}$ 
are distance distribution functions.
In addition^{} to ${F}_{(a,b)}$ being a distance distribution function, we need that ${F}_{(a,b)}={e}_{0}$ 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 ${\mathrm{\Delta}}^{+}$ by $F\le G$ iff $F(x)\le G(x)$ for all $x\in \mathbb{R}$. It is not hard to see that ${e}_{x}\le {e}_{y}$ iff $y\le x$ and that ${e}_{0}$ is the top element of ${\mathrm{\Delta}}^{+}$. From the poset ${\mathrm{\Delta}}^{+}$, call a binary operator $\tau $ on ${\mathrm{\Delta}}^{+}$ a triangle function if $\tau $ turns ${\mathrm{\Delta}}^{+}$ into a partially ordered (http://planetmath.org/PartiallyOrderedGroup) commutative monoid with ${e}_{0}$ serving as the identity element^{}. Spelling this out, for any $F,G,H\in {\mathrm{\Delta}}^{+}$, we have

•
$F\tau G=G\tau F$,

•
$(F\tau G)\tau H=F\tau (G\tau H)$,

•
$F\tau {e}_{0}={e}_{0}\tau F=F$, and

•
if $G\le H$, then $F\tau G\le F\tau H$,
where $F\tau G$ means $\tau (F,G)$. For example, $F\tau G=F\cdot G$, $F\tau G=\mathrm{min}(F,G)$ are two triangle functions. In fact, since $F\tau G\le F\tau {e}_{0}=F$ and $F\tau G\le G$ similarly, we have $F\tau G\le \mathrm{min}(F,G)$ for any triangle function $\tau $.
With this, we are ready for our main definition:
Definition. A probabilistic metric space is a (nonempty) set $X$, equipped with a function $F:X\times X\to {\mathrm{\Delta}}^{+}$, where ${\mathrm{\Delta}}^{+}$ is the set of distance distribution functions on which a triangle function $\tau $ is defined, such that

1.
${F}_{(a,b)}={e}_{0}$ iff $a=b$, where ${F}_{(a,b)}:=F(a,b)$,

2.
${F}_{(a,b)}={F}_{(b,a)}$, and

3.
${F}_{(a,c)}\ge {F}_{(a,b)}\tau {F}_{(b,c)}$.
Given a metric space $(X,d)$, if we can find a triangle function $\tau $ such that ${e}_{x}\tau {e}_{y}={e}_{x+y}$, then $(X,F)$ with ${F}_{(a,b)}:={e}_{d(a,b)}$ is a probabilistic metric space.
References
 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, 320, (1962).
Title  probabilistic metric space 

Canonical name  ProbabilisticMetricSpace 
Date of creation  20130322 16:49:38 
Last modified on  20130322 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 