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 [a1].
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
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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 [a2]. 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 [a3], [a4].
References
[a1] | B. Schweizer, A. Sklar, "Probabilistic metric spaces" , Elsevier & North-Holland (1983) MR0790314 MR0719088 Zbl 0546.60010 Zbl 0524.54005 |
[a2] | M.F. Janowitz, B. Schweizer, "Ordinal and percentile clustering" Math. Social Sci. , 18 (1989) pp. 135–186 MR1016504 Zbl 0695.62144 |
[a3] | 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 |
[a4] | B. Schweizer, A. Sklar, J. Smítal, "Distributional (and other) chaos and its measurement" (to appear) |
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