Probabilistic metric space

2020 Mathematics Subject Classification: Primary: 54E70 [MSN][ZBL]

Generalizations of metric spaces (cf. Metric space), in which the distances between points are specified by probability distributions (cf. Probability distribution) rather than numbers. The general notion was introduced by K. Menger in 1942 and has since been developed by a number of authors. A treatment, comprehensive up to 1983, may be found in [SSk].

Let be the set of all functions from the real line into the unit interval = that are non-decreasing and left-continuous on , and such that and , i.e., the set of all probability distribution functions whose support lies in the extended half-line . For any , let be defined by for and for ; and let be defined by for all and . Then, under the usual pointwise ordering of functions, given by if and only if for all , the set is a complete lattice with maximal element and minimal element . There is a natural topological structure (topology) on , namely, the topology of weak convergence (cf. also Weak topology), where if and only if at every point of continuity of . Under this topology is compact and connected (cf. Compact space; Connected space); moreover, this topology can be metrized (cf. Metrizable space), e.g., by a variant of the Lévy metric.

A triangle function is a binary operation on satisfying the following conditions:

a) for all ;

b) whenever , ;

c) ;

d) .

It is also often required that be continuous with respect to the topology of weak convergence, or that satisfies the condition:

e) for all .

Examples of triangle functions are convolution and the functions given by

Here is a -norm, i.e., a binary operation on that, like , has an identity element (the number in this case) and is non-decreasing, commutative, and associative. Particular -norms are the functions , , and given, respectively, by , , and . The corresponding triangle functions , , and are continuous and satisfy e).

A probabilistic metric space is a triple , where is a set, is a function from into , is a triangle function, such that for any ,

I) ;

II) if ;

III) ;

IV) .

If satisfies only I), III) and IV), then is a probabilistic pseudo-metric space.

For any and any , the value of at , usually denoted by , is often interpreted as "the probability that the distance between p and q is less than x" .

Thus, the generalization from ordinary to probabilistic metric spaces consists of:

1) replacing the range of the ordinary metric by the space of probability distributions ;

2) replacing the operation of addition on , which plays the pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function from into , if is defined via and if is a triangle function satisfying e), then is an ordinary metric space; and conversely. If for some -norm , then the probabilistic metric space is a Menger space.

There is a natural topology on a probabilistic metric space, determined by the system of neighbourhoods . However, a more interesting class of topological structures is obtained by designating a particular as a profile function, interpreting as the maximum confidence associated with distances less than , and considering the system of neighbourhoods

These determine a generalized topology (specifically, a closure space in the sense of E. Čech). There is also an associated indistinguishability relation, defined by if and only if . This relation is a tolerance relation, i.e., is reflexive and symmetric, but not necessarily transitive.

Let be a probability space, a metric space, and the set of all functions from into . For any , define via

Then IV) holds with . The resultant probabilistic pseudo-metric space is called an -space. For any , the function from into given by is a pseudo-metric on , and the -space is pseudo-metrically generated in the sense that

Conversely, any such pseudo-metrically generated space is an -space. An important class of -spaces is obtained when is the Euclidean -dimensional space and is the set of all non-degenerate -dimensional spherically symmetric Gaussian vectors.

The idea behind the construction of an -space has been generalized. For example, if is a set with some structure, e.g., a normed, inner product or topological space, then the set of all functions from into yields a space in which that structure is probabilistic. This idea has recently been applied in cluster analysis, where the numerical dissimilarity coefficient has been replaced by an element of . The result is a theory of percentile clustering [JS]. The principal advantage of percentile clustering methods is that, when working with distributed data, they permit one to classify first and then summarize, instead of summarizing first and then classifying.

Let be a function from a metric space into itself, and, for any non-negative integer , let denote the th iterate of . For any , define the sequence by

and for any positive integer define via

where denotes the cardinality of the set in question. The number may be interpreted as the probability that the distance between the initial segments, of length , of the trajectories of and is less than . Let

and let and be normalized to be left-continuous, hence in . If is defined via , then, again, IV) holds with . The resultant probabilistic pseudo-metric space is a transformation generated space. Note that , so that is (probabilistic) distance-preserving.

If is measure-preserving (cf. Measure-preserving transformation) with respect to a probability measure on , then for almost all pairs in ; and if, in addition, is mixing, then there is a such that for almost all pairs .

The above ideas play an important role in chaos theory. For example, if is a closed interval , if is continuous, and if there is a single pair of points for which , then is chaotic in a very strong sense. This fact leads to a theory of distributional chaos. Specifically, if is compact, then is distributionally chaotic if and only if there is a pair of points for which . Furthermore, the number

where is the diameter of , provides a useful measure of the degree of distributional chaos. For details see [SSm], [SSS].

References

[SSk]	B. Schweizer, A. Sklar, "Probabilistic metric spaces" , Elsevier & North-Holland (1983) MR0790314 MR0719088 Zbl 0546.60010 Zbl 0524.54005
[JS]	M.F. Janowitz, B. Schweizer, "Ordinal and percentile clustering" Math. Social Sci. , 18 (1989) pp. 135–186 MR1016504 Zbl 0695.62144
[SSm]	B. Schweizer, J. Smítal, "Measures of chaos and a spectral decomposition of dynamical systems on the interval" Trans. Amer. Math. Soc. , 344 (1994) pp. 737–754 MR1227094 Zbl 0812.58062
[SSS]	B. Schweizer, A. Sklar, J. Smítal, "Distributional (and other) chaos and its measurement" (to appear)