A probabilistic metric space is a generalization of metric spaces where the distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

 is no longer defined on positive real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, but on distribution functions.

Let D+ be the set of all probability distribution function

Probability distribution function

Depending upon which text is consulted, a probability distribution function is any of:* a probability distribution function,* a cumulative distribution function,* a probability mass function, or* a probability density function....

s F such that F(0) = 0: F is a nondecreasing, left continuous mapping from the real numbers R into [0, 1] such that

sup F(x) = 1

where the supremum

Supremum

In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

 is taken over all x in R.

The ordered pair

Ordered pair

In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

 (S,d) is said to be a probabilistic metric space if S is a nonempty set and

d: S×S →D+

In the following, d(p, q) is denoted by d_p,q for every (p, q) ∈ S × S and is a distribution function d_p,q(x). The distance-distribution function satisfies the following conditions:

• d_u,v(x) = 0 for all x > 0 ⇔ u = v (u, v ∈ S).
• d_u,v(x) = d_v,u(x) for all x and for every u, v ∈ S.
• d_u,v(x) = 1 and d_v,w(y) = 1 ⇒ d_u,w(x + y) = 1 for u, v, w ∈ S and x, y ∈ R.

See also

• Statistical distance
  Statistical distance
  In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two samples, two random variables, or two probability distributions, for example.-Metrics:...