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Recall that a metric space is a set $X$ equipped with a distance function $d:X\times X\to [0,\infty)$ , such that
- $d(a,b)=0$ iff $a=b$ ,
- $d(a,b)=d(b,a)$ , and
- $d(a,c)\le d(a,b)+d(b,c)$ .
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 $\Delta^+$ . For example, for any $r\ge 0$ , the step functions defined by \begin{eqnarray*} e_r(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x\le r, \\ 1 & \mbox{otherwise} \\ \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 $\Delta^+$ 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 $\Delta^+$ . From the poset $\Delta^+$ , call a binary operator $\tau$ on $\Delta^+$ a triangle
function if $\tau$ turns $\Delta^+$ into a partially ordered 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 $G\le H$ , 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 $F:X\times X\to \Delta^+$ , where $\Delta^+$ is the set of distance distribution functions on which a triangle function $\tau$ is defined, such that
- $F_{(a,b)}=e_0$ iff $a=b$ , where $F_{(a,b)}:=F(a,b)$ ,
- $F_{(a,b)}=F_{(b,a)}$ , and
- $F_{(a,c)}\ge F_{(a,b)}\tau F_{(b,c)}$ .
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.
- 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. Ucen. Zap. 122, 3-20, (1962).
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