# probabilistic metric space

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

1. 1.

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

2. 2.

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

3. 3.

$d(a,c)\leq d(a,b)+d(b,c)$.

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 $\Delta^{+}$. For example, for any $r\geq 0$, the step functions defined by

 $\displaystyle e_{r}(x)$ $\displaystyle=$ $\displaystyle\left\{\begin{array}[]{ll}0&\mbox{when}\,\,x\leq r,\\ 1&\mbox{otherwise}\\ \end{array}\right.$

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 $\Delta^{+}$ by $F\leq G$ iff $F(x)\leq G(x)$ for all $x\in\mathbb{R}$. It is not hard to see that $e_{x}\leq e_{y}$ iff $y\leq x$ and that $e_{0}$ is the top element of $\Delta^{+}$. From the poset $\Delta^{+}$, call a binary operator $\tau$ on $\Delta^{+}$ a triangle function if $\tau$ turns $\Delta^{+}$ into a partially ordered (http://planetmath.org/PartiallyOrderedGroup) commutative monoid with $e_{0}$ serving as the identity element. Spelling this out, for any $F,G,H\in\Delta^{+}$, 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 $G\leq H$, then $F\tau G\leq F\tau H$,

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 $F\tau G\leq F\tau e_{0}=F$ and $F\tau G\leq G$ similarly, we have $F\tau G\leq\min(F,G)$ 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 $F:X\times X\to\Delta^{+}$, where $\Delta^{+}$ is the set of distance distribution functions on which a triangle function $\tau$ is defined, such that

1. 1.

$F_{(a,b)}=e_{0}$ iff $a=b$, where $F_{(a,b)}:=F(a,b)$,

2. 2.

$F_{(a,b)}=F_{(b,a)}$, and

3. 3.

$F_{(a,c)}\geq F_{(a,b)}\tau F_{(b,c)}$.

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, 3-20, (1962).
Title probabilistic metric space ProbabilisticMetricSpace 2013-03-22 16:49:38 2013-03-22 16:49:38 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 54E70 distance distribution function triangle function