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

$e_r(x)$ $=$

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 $x\in ℝ$. 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, $F\tau G=F\cdot G$, $F\tau G=\min (F,G)$ 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 $F\tau G\le \min (F,G)$ for any triangle function $\tau$.

With this, we are ready for our main definition:

Definition. A probabilistic metric space is a (non-empty) set $X$, equipped with a function $F:X\times X\to {\mathrm{\Delta }}^{+}$, 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.

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