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Generalizations of metric spaces (cf. [[Metric space|Metric space]]), in which the distances between points are specified by probability distributions (cf. [[Probability distribution|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 {{Cite|SSk}}.
 
Generalizations of metric spaces (cf. [[Metric space|Metric space]]), in which the distances between points are specified by probability distributions (cf. [[Probability distribution|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 {{Cite|SSk}}.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102301.png" /> be the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102302.png" /> from the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102303.png" /> into the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102304.png" /> =
+
Let $  \Delta  ^ {+} $
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102305.png" /> that are non-decreasing and left-continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102306.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102308.png" />, i.e., the set of all probability distribution functions whose support lies in the extended half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p1102309.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023010.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023011.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023015.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023016.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023019.png" />. Then, under the usual pointwise ordering of functions, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023020.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023022.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023023.png" /> is a [[Complete lattice|complete lattice]] with maximal element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023024.png" /> and minimal element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023025.png" />. There is a natural [[Topological structure (topology)|topological structure (topology)]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023026.png" />, namely, the topology of weak convergence (cf. also [[Weak topology|Weak topology]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023028.png" /> at every point of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023029.png" />. Under this topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023030.png" /> is compact and connected (cf. [[Compact space|Compact space]]; [[Connected space|Connected space]]); moreover, this topology can be metrized (cf. [[Metrizable space|Metrizable space]]), e.g., by a variant of the [[Lévy metric|Lévy metric]].
+
be the set of all functions $  F $
 +
from the real line $  \mathbf R $
 +
into the unit interval $  I $
 +
=
 +
$  [ 0,1 ] $
 +
that are non-decreasing and left-continuous on $  [ 0, \infty ) $,
 +
and such that $  F ( 0 ) = 0 $
 +
and $  F ( \infty ) = 1 $,  
 +
i.e., the set of all probability distribution functions whose support lies in the extended half-line $  \mathbf R  ^ {+} = [ 0, \infty ] $.  
 +
For any $  a \in [ 0, \infty ) $,
 +
let $  \epsilon _ {a} \in \Delta  ^ {+} $
 +
be defined by $  \epsilon _ {a} ( x ) = 0 $
 +
for $  x \leq  a $
 +
and $  \epsilon _ {a} ( x ) = 1 $
 +
for $  x > a $;  
 +
and let $  \epsilon _  \infty  \in \Delta  ^ {+} $
 +
be defined by $  \epsilon _  \infty  ( x ) = 0 $
 +
for all $  x < \infty $
 +
and $  \epsilon _  \infty  ( \infty ) = 1 $.  
 +
Then, under the usual pointwise ordering of functions, given by $  F \leq  G $
 +
if and only if $  F ( x ) \leq  G ( x ) $
 +
for all $  x \in \mathbf R $,  
 +
the set $  \Delta  ^ {+} $
 +
is a [[Complete lattice|complete lattice]] with maximal element $  \epsilon _ {0} $
 +
and minimal element $  \epsilon _  \infty  $.  
 +
There is a natural [[Topological structure (topology)|topological structure (topology)]] on $  \Delta  ^ {+} $,  
 +
namely, the topology of weak convergence (cf. also [[Weak topology|Weak topology]]), where $  F _ {n} \rightarrow F $
 +
if and only if $  F _ {n} ( x ) \rightarrow F ( x ) $
 +
at every point of continuity of $  F $.  
 +
Under this topology $  \Delta  ^ {+} $
 +
is compact and connected (cf. [[Compact space|Compact space]]; [[Connected space|Connected space]]); moreover, this topology can be metrized (cf. [[Metrizable space|Metrizable space]]), e.g., by a variant of the [[Lévy metric|Lévy metric]].
  
A triangle function is a binary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023032.png" /> satisfying the following conditions:
+
A triangle function is a binary operation $  \tau $
 +
on $  \Delta  ^ {+} $
 +
satisfying the following conditions:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023034.png" />;
+
a) $  \tau ( F, \epsilon _ {0} ) = F $
 +
for all $  F \in \Delta  ^ {+} $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023035.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023037.png" />;
+
b) $  \tau ( E,F ) \leq  \tau ( G,H ) $
 +
whenever $  E \leq  G $,  
 +
$  F \leq  H $;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023038.png" />;
+
c) $  \tau ( E,F ) = \tau ( F,E ) $;
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023039.png" />.
+
d) $  \tau ( E, \tau ( F,G ) ) = \tau ( \tau ( E,F ) ,G ) $.
  
It is also often required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023040.png" /> be continuous with respect to the topology of weak convergence, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023041.png" /> satisfies the condition:
+
It is also often required that $  \tau $
 +
be continuous with respect to the topology of weak convergence, or that $  \tau $
 +
satisfies the condition:
  
e) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023043.png" />.
+
e) $  \tau ( \epsilon _ {a} , \epsilon _ {b} ) \geq  \epsilon _ {a + b }  $
 +
for all $  a,b \in \mathbf R  ^ {+} $.
  
Examples of triangle functions are convolution and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023044.png" /> given by
+
Examples of triangle functions are convolution and the functions $  \tau _ {T} $
 +
given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023045.png" /></td> </tr></table>
+
$$
 +
\tau _ {T} ( F,G ) ( x ) = \sup  _ {u + v = x } T ( F ( u ) ,G ( v ) ) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023046.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023048.png" />-norm, i.e., a binary operation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023049.png" /> that, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023050.png" />, has an identity element (the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023051.png" /> in this case) and is non-decreasing, commutative, and associative. Particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023052.png" />-norms are the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023054.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023055.png" /> given, respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023057.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023058.png" />. The corresponding triangle functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023061.png" /> are continuous and satisfy e).
+
Here $  T $
 +
is a $  t $-
 +
norm, i.e., a binary operation on $  I $
 +
that, like $  \tau $,  
 +
has an identity element (the number $  1 $
 +
in this case) and is non-decreasing, commutative, and associative. Particular $  t $-
 +
norms are the functions $  W $,  
 +
$  \Pi $,  
 +
and $  M $
 +
given, respectively, by $  W ( a,b ) = \max  ( a + b - 1,0 ) $,
 +
$  \Pi ( a,b ) = ab $,  
 +
and $  M ( a,b ) = \min  ( a,b ) $.  
 +
The corresponding triangle functions $  \tau _ {W} $,  
 +
$  \tau _  \Pi  $,  
 +
and $  \tau _ {M} $
 +
are continuous and satisfy e).
  
A probabilistic metric space is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023063.png" /> is a set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023064.png" /> is a function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023065.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023067.png" /> is a triangle function, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023068.png" />,
+
A probabilistic metric space is a triple $  ( S, {\mathcal F}, \tau ) $,  
 +
where $  S $
 +
is a set, $  {\mathcal F} $
 +
is a function from $  S \times S $
 +
into $  \Delta  ^ {+} $,  
 +
$  \tau $
 +
is a triangle function, such that for any p,q,r \in S $,
  
I) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023069.png" />;
+
I) $  {\mathcal F} ( p,p ) = \epsilon _ {0} $;
  
II) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023070.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023071.png" />;
+
II) $  {\mathcal F} ( p,q ) \neq \epsilon _ {0} $
 +
if p \neq q $;
  
III) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023072.png" />;
+
III) $  {\mathcal F} ( p,q ) = {\mathcal F} ( q,p ) $;
  
IV) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023073.png" />.
+
IV) $  {\mathcal F} ( p,r ) \geq  \tau ( {\mathcal F} ( p,q ) , {\mathcal F} ( q,r ) ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023074.png" /> satisfies only I), III) and IV), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023075.png" /> is a probabilistic pseudo-metric space.
+
If $  {\mathcal F} $
 +
satisfies only I), III) and IV), then $  ( S, {\mathcal F}, \tau ) $
 +
is a probabilistic pseudo-metric space.
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023076.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023077.png" />, the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023078.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023079.png" />, usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023080.png" />, is often interpreted as "the probability that the distance between p and q is less than x" .
+
For any $  x \in \mathbf R  ^ {+} $
 +
and any p,q \in S $,
 +
the value of $  {\mathcal F} ( p,q ) $
 +
at $  x $,  
 +
usually denoted by $  F _ {pq }  ( x ) $,  
 +
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:
 
Thus, the generalization from ordinary to probabilistic metric spaces consists of:
  
1) replacing the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023081.png" /> of the ordinary metric by the space of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023082.png" />;
+
1) replacing the range $  \mathbf R  ^ {+} $
 +
of the ordinary metric by the space of probability distributions $  \Delta  ^ {+} $;
  
2) replacing the operation of addition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023083.png" />, which plays the pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023084.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023085.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023086.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023087.png" /> is defined via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023088.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023089.png" /> is a triangle function satisfying e), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023090.png" /> is an ordinary metric space; and conversely. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023091.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023092.png" />-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023093.png" />, then the probabilistic metric space is a Menger space.
+
2) replacing the operation of addition on $  \mathbf R  ^ {+} $,  
 +
which plays the pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function $  d $
 +
from $  S \times S $
 +
into $  \mathbf R  ^ {+} $,  
 +
if $  {\mathcal F} $
 +
is defined via $  F _ {pq }  = \epsilon _ {d ( p,q ) }  $
 +
and if $  \tau $
 +
is a triangle function satisfying e), then $  ( S,d ) $
 +
is an ordinary metric space; and conversely. If $  \tau = \tau _ {T} $
 +
for some $  t $-
 +
norm $  T $,  
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023094.png" />. However, a more interesting class of topological structures is obtained by designating a particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023095.png" /> as a profile function, interpreting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023096.png" /> as the maximum confidence associated with distances less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023097.png" />, and considering the system of neighbourhoods
+
There is a natural topology on a probabilistic metric space, determined by the system of neighbourhoods $  N _ {p} ( \epsilon, \delta ) = \{ {q \in S } : {F _ {pq }  ( \epsilon ) > 1 - \delta } \} $.  
 +
However, a more interesting class of topological structures is obtained by designating a particular $  \phi \in \Delta  ^ {+} $
 +
as a profile function, interpreting $  \phi ( x ) $
 +
as the maximum confidence associated with distances less than $  x $,  
 +
and considering the system of neighbourhoods
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023098.png" /></td> </tr></table>
+
$$
 +
N _ {p} ( \phi, \epsilon ) = \left \{ q : {F _ {pq }  ( x + \epsilon ) \geq  \phi ( x ) - \epsilon ,  x \in ( 0, {1 / \epsilon } ) } \right \} .
 +
$$
  
These determine a generalized topology (specifically, a [[Closure space|closure space]] in the sense of E. Čech). There is also an associated indistinguishability relation, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p11023099.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230100.png" />. This relation is a tolerance relation, i.e., is reflexive and symmetric, but not necessarily transitive.
+
These determine a generalized topology (specifically, a [[Closure space|closure space]] in the sense of E. Čech). There is also an associated indistinguishability relation, defined by p ( { \mathop{\rm ind} } \phi ) q $
 +
if and only if $  F _ {pq }  \geq  \phi $.  
 +
This relation is a tolerance relation, i.e., is reflexive and symmetric, but not necessarily transitive.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230101.png" /> be a probability space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230102.png" /> a metric space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230103.png" /> the set of all functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230104.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230105.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230106.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230107.png" /> via
+
Let $  ( \Omega, {\mathcal A},P ) $
 +
be a probability space, $  ( M,d ) $
 +
a metric space, and $  S $
 +
the set of all functions from $  \Omega $
 +
into $  M $.  
 +
For any p,q \in S $,
 +
define $  {\mathcal F} ( p,q ) $
 +
via
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230108.png" /></td> </tr></table>
+
$$
 +
F _ {pq }  ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d ( p ( \omega ) ,q ( \omega ) ) < t } \right \} .
 +
$$
  
Then IV) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230109.png" />. The resultant probabilistic pseudo-metric space is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230111.png" />-space. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230112.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230113.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230114.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230115.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230116.png" /> is a [[Pseudo-metric|pseudo-metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230117.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230118.png" />-space is pseudo-metrically generated in the sense that
+
Then IV) holds with $  \tau = \tau _ {W} $.  
 +
The resultant probabilistic pseudo-metric space is called an $  E $-
 +
space. For any $  \omega \in \Omega $,  
 +
the function $  d _  \omega  $
 +
from $  S \times S $
 +
into $  \mathbf R  ^ {+} $
 +
given by $  d _  \omega  ( p,q ) = d ( p ( \omega ) ,q ( \omega ) ) $
 +
is a [[Pseudo-metric|pseudo-metric]] on $  S $,  
 +
and the $  E $-
 +
space is pseudo-metrically generated in the sense that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230119.png" /></td> </tr></table>
+
$$
 +
F _ {pq }  ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d _  \omega  ( p,q ) < t } \right \} .
 +
$$
  
Conversely, any such pseudo-metrically generated space is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230120.png" />-space. An important class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230121.png" />-spaces is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230122.png" /> is the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230123.png" />-dimensional space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230124.png" /> is the set of all non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230125.png" />-dimensional spherically symmetric Gaussian vectors.
+
Conversely, any such pseudo-metrically generated space is an $  E $-
 +
space. An important class of $  E $-
 +
spaces is obtained when $  ( M,d ) $
 +
is the Euclidean $  n $-
 +
dimensional space and $  S $
 +
is the set of all non-degenerate $  n $-
 +
dimensional spherically symmetric Gaussian vectors.
  
The idea behind the construction of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230126.png" />-space has been generalized. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230127.png" /> is a set with some structure, e.g., a normed, inner product or topological space, then the set of all functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230128.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230129.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230130.png" />. The result is a theory of percentile clustering {{Cite|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.
+
The idea behind the construction of an $  E $-
 +
space has been generalized. For example, if $  \Sigma $
 +
is a set with some structure, e.g., a normed, inner product or topological space, then the set of all functions from $  ( \Omega, {\mathcal A}, {\mathsf P} ) $
 +
into $  \Sigma $
 +
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 $  \Delta  ^ {+} $.  
 +
The result is a theory of percentile clustering {{Cite|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230131.png" /> be a function from a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230132.png" /> into itself, and, for any non-negative integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230133.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230134.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230135.png" />th iterate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230136.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230137.png" />, define the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230138.png" /> by
+
Let $  f $
 +
be a function from a [[Metric space|metric space]] $  ( S,d ) $
 +
into itself, and, for any non-negative integer $  m $,  
 +
let $  f  ^ {m} $
 +
denote the $  m $
 +
th iterate of $  f $.  
 +
For any p,q \in S $,
 +
define the sequence $  \delta _ {pq }  $
 +
by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230139.png" /></td> </tr></table>
+
$$
 +
\delta _ {pq }  ( m ) = d ( f  ^ {m} ( p ) ,f  ^ {m} ( q ) ) ,
 +
$$
  
and for any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230140.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230141.png" /> via
+
and for any positive integer $  n $
 +
define $  F _ {pq }  ^ {( n ) } \in \Delta  ^ {+} $
 +
via
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230142.png" /></td> </tr></table>
+
$$
 +
F _ {pq }  ^ {( n ) } ( t ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230143.png" /></td> </tr></table>
+
$$
 +
=  
 +
{
 +
\frac{1}{n}
 +
}  \# \left \{ m : {0 \leq  m \leq  n - 1,  \delta _ {m} ( p,q ) < t } \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230144.png" /> denotes the cardinality of the set in question. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230145.png" /> may be interpreted as the probability that the distance between the initial segments, of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230146.png" />, of the trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230148.png" /> is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230149.png" />. Let
+
where $  \# $
 +
denotes the cardinality of the set in question. The number $  F _ {pq }  ^ {( n ) } ( t ) $
 +
may be interpreted as the probability that the distance between the initial segments, of length $  n $,  
 +
of the trajectories of p $
 +
and $  q $
 +
is less than $  t $.  
 +
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230150.png" /></td> </tr></table>
+
$$
 +
F _ {pq }  ( t ) = {\lim\limits  \inf } _ {n \rightarrow \infty } F _ {pq }  ^ {( n ) } ( t ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230151.png" /></td> </tr></table>
+
$$
 +
F _ {pq }  ^ {*} ( t ) = {\lim\limits  \sup } _ {n \rightarrow \infty } F _ {pq }  ^ {( n ) } ( t ) ,
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230153.png" /> be normalized to be left-continuous, hence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230154.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230155.png" /> is defined via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230156.png" />, then, again, IV) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230157.png" />. The resultant probabilistic pseudo-metric space is a transformation generated space. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230158.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230159.png" /> is (probabilistic) distance-preserving.
+
and let $  F _ {pq }  $
 +
and $  F _ {pq }  ^ {*} $
 +
be normalized to be left-continuous, hence in $  \Delta  ^ {+} $.  
 +
If $  {\mathcal F} $
 +
is defined via $  {\mathcal F} ( p,q ) = F _ {pq }  $,  
 +
then, again, IV) holds with $  \tau = \tau _ {W} $.  
 +
The resultant probabilistic pseudo-metric space is a transformation generated space. Note that $  F _ {pq }  = F _ {f ( p ) f ( q ) }  $,  
 +
so that $  f $
 +
is (probabilistic) distance-preserving.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230160.png" /> is measure-preserving (cf. [[Measure-preserving transformation|Measure-preserving transformation]]) with respect to a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230161.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230162.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230163.png" /> for almost all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230164.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230165.png" />; and if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230166.png" /> is [[Mixing|mixing]], then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230167.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230168.png" /> for almost all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230169.png" />.
+
If $  f $
 +
is measure-preserving (cf. [[Measure-preserving transformation|Measure-preserving transformation]]) with respect to a [[Probability measure|probability measure]] $  {\mathsf P} $
 +
on $  S $,  
 +
then $  F _ {pq }  = F _ {pq }  ^ {*} $
 +
for almost all pairs $  ( p,q ) $
 +
in $  S \times S $;  
 +
and if, in addition, $  f $
 +
is [[Mixing|mixing]], then there is a $  G \in \Delta  ^ {+} $
 +
such that $  F _ {pq }  = F _ {pq }  ^ {*} = G $
 +
for almost all pairs $  ( p,q ) $.
  
The above ideas play an important role in chaos theory. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230170.png" /> is a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230171.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230172.png" /> is continuous, and if there is a single pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230173.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230174.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230175.png" /> is chaotic in a very strong sense. This fact leads to a theory of distributional chaos. Specifically, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230176.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230177.png" /> is distributionally chaotic if and only if there is a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230178.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230179.png" />. Furthermore, the number
+
The above ideas play an important role in chaos theory. For example, if $  S $
 +
is a closed interval $  [ a,b ] $,  
 +
if $  f $
 +
is continuous, and if there is a single pair of points p,q \in [ a,b ] $
 +
for which $  F _ {pq }  \neq F _ {pq }  ^ {*} $,  
 +
then $  f $
 +
is chaotic in a very strong sense. This fact leads to a theory of distributional chaos. Specifically, if $  ( S,d ) $
 +
is compact, then $  f $
 +
is distributionally chaotic if and only if there is a pair of points p,q \in S $
 +
for which $  F _ {pq }  \neq F _ {pq }  ^ {*} $.  
 +
Furthermore, the number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230180.png" /></td> </tr></table>
+
$$
 +
\mu ( f ) = \sup  _ {p,q \in S } {
 +
\frac{1}{d _ {S} }
 +
} \int\limits _ { 0 } ^  \infty  {( F _ {pq }  ^ {*} ( t ) - F _ {pq }  ( t ) ) }  {dt } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230181.png" /> is the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110230/p110230182.png" />, provides a useful measure of the degree of distributional chaos. For details see {{Cite|SSm}}, {{Cite|SSS}}.
+
where $  d _ {S} $
 +
is the diameter of $  S $,  
 +
provides a useful measure of the degree of distributional chaos. For details see {{Cite|SSm}}, {{Cite|SSS}}.
  
 
====References====
 
====References====

Latest revision as of 08:07, 6 June 2020


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 $ \Delta ^ {+} $ be the set of all functions $ F $ from the real line $ \mathbf R $ into the unit interval $ I $ = $ [ 0,1 ] $ that are non-decreasing and left-continuous on $ [ 0, \infty ) $, and such that $ F ( 0 ) = 0 $ and $ F ( \infty ) = 1 $, i.e., the set of all probability distribution functions whose support lies in the extended half-line $ \mathbf R ^ {+} = [ 0, \infty ] $. For any $ a \in [ 0, \infty ) $, let $ \epsilon _ {a} \in \Delta ^ {+} $ be defined by $ \epsilon _ {a} ( x ) = 0 $ for $ x \leq a $ and $ \epsilon _ {a} ( x ) = 1 $ for $ x > a $; and let $ \epsilon _ \infty \in \Delta ^ {+} $ be defined by $ \epsilon _ \infty ( x ) = 0 $ for all $ x < \infty $ and $ \epsilon _ \infty ( \infty ) = 1 $. Then, under the usual pointwise ordering of functions, given by $ F \leq G $ if and only if $ F ( x ) \leq G ( x ) $ for all $ x \in \mathbf R $, the set $ \Delta ^ {+} $ is a complete lattice with maximal element $ \epsilon _ {0} $ and minimal element $ \epsilon _ \infty $. There is a natural topological structure (topology) on $ \Delta ^ {+} $, namely, the topology of weak convergence (cf. also Weak topology), where $ F _ {n} \rightarrow F $ if and only if $ F _ {n} ( x ) \rightarrow F ( x ) $ at every point of continuity of $ F $. Under this topology $ \Delta ^ {+} $ 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 $ \tau $ on $ \Delta ^ {+} $ satisfying the following conditions:

a) $ \tau ( F, \epsilon _ {0} ) = F $ for all $ F \in \Delta ^ {+} $;

b) $ \tau ( E,F ) \leq \tau ( G,H ) $ whenever $ E \leq G $, $ F \leq H $;

c) $ \tau ( E,F ) = \tau ( F,E ) $;

d) $ \tau ( E, \tau ( F,G ) ) = \tau ( \tau ( E,F ) ,G ) $.

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

e) $ \tau ( \epsilon _ {a} , \epsilon _ {b} ) \geq \epsilon _ {a + b } $ for all $ a,b \in \mathbf R ^ {+} $.

Examples of triangle functions are convolution and the functions $ \tau _ {T} $ given by

$$ \tau _ {T} ( F,G ) ( x ) = \sup _ {u + v = x } T ( F ( u ) ,G ( v ) ) . $$

Here $ T $ is a $ t $- norm, i.e., a binary operation on $ I $ that, like $ \tau $, has an identity element (the number $ 1 $ in this case) and is non-decreasing, commutative, and associative. Particular $ t $- norms are the functions $ W $, $ \Pi $, and $ M $ given, respectively, by $ W ( a,b ) = \max ( a + b - 1,0 ) $, $ \Pi ( a,b ) = ab $, and $ M ( a,b ) = \min ( a,b ) $. The corresponding triangle functions $ \tau _ {W} $, $ \tau _ \Pi $, and $ \tau _ {M} $ are continuous and satisfy e).

A probabilistic metric space is a triple $ ( S, {\mathcal F}, \tau ) $, where $ S $ is a set, $ {\mathcal F} $ is a function from $ S \times S $ into $ \Delta ^ {+} $, $ \tau $ is a triangle function, such that for any $ p,q,r \in S $,

I) $ {\mathcal F} ( p,p ) = \epsilon _ {0} $;

II) $ {\mathcal F} ( p,q ) \neq \epsilon _ {0} $ if $ p \neq q $;

III) $ {\mathcal F} ( p,q ) = {\mathcal F} ( q,p ) $;

IV) $ {\mathcal F} ( p,r ) \geq \tau ( {\mathcal F} ( p,q ) , {\mathcal F} ( q,r ) ) $.

If $ {\mathcal F} $ satisfies only I), III) and IV), then $ ( S, {\mathcal F}, \tau ) $ is a probabilistic pseudo-metric space.

For any $ x \in \mathbf R ^ {+} $ and any $ p,q \in S $, the value of $ {\mathcal F} ( p,q ) $ at $ x $, usually denoted by $ F _ {pq } ( x ) $, 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 $ \mathbf R ^ {+} $ of the ordinary metric by the space of probability distributions $ \Delta ^ {+} $;

2) replacing the operation of addition on $ \mathbf R ^ {+} $, which plays the pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function $ d $ from $ S \times S $ into $ \mathbf R ^ {+} $, if $ {\mathcal F} $ is defined via $ F _ {pq } = \epsilon _ {d ( p,q ) } $ and if $ \tau $ is a triangle function satisfying e), then $ ( S,d ) $ is an ordinary metric space; and conversely. If $ \tau = \tau _ {T} $ for some $ t $- norm $ T $, 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 $ N _ {p} ( \epsilon, \delta ) = \{ {q \in S } : {F _ {pq } ( \epsilon ) > 1 - \delta } \} $. However, a more interesting class of topological structures is obtained by designating a particular $ \phi \in \Delta ^ {+} $ as a profile function, interpreting $ \phi ( x ) $ as the maximum confidence associated with distances less than $ x $, and considering the system of neighbourhoods

$$ N _ {p} ( \phi, \epsilon ) = \left \{ q : {F _ {pq } ( x + \epsilon ) \geq \phi ( x ) - \epsilon , x \in ( 0, {1 / \epsilon } ) } \right \} . $$

These determine a generalized topology (specifically, a closure space in the sense of E. Čech). There is also an associated indistinguishability relation, defined by $ p ( { \mathop{\rm ind} } \phi ) q $ if and only if $ F _ {pq } \geq \phi $. This relation is a tolerance relation, i.e., is reflexive and symmetric, but not necessarily transitive.

Let $ ( \Omega, {\mathcal A},P ) $ be a probability space, $ ( M,d ) $ a metric space, and $ S $ the set of all functions from $ \Omega $ into $ M $. For any $ p,q \in S $, define $ {\mathcal F} ( p,q ) $ via

$$ F _ {pq } ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d ( p ( \omega ) ,q ( \omega ) ) < t } \right \} . $$

Then IV) holds with $ \tau = \tau _ {W} $. The resultant probabilistic pseudo-metric space is called an $ E $- space. For any $ \omega \in \Omega $, the function $ d _ \omega $ from $ S \times S $ into $ \mathbf R ^ {+} $ given by $ d _ \omega ( p,q ) = d ( p ( \omega ) ,q ( \omega ) ) $ is a pseudo-metric on $ S $, and the $ E $- space is pseudo-metrically generated in the sense that

$$ F _ {pq } ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d _ \omega ( p,q ) < t } \right \} . $$

Conversely, any such pseudo-metrically generated space is an $ E $- space. An important class of $ E $- spaces is obtained when $ ( M,d ) $ is the Euclidean $ n $- dimensional space and $ S $ is the set of all non-degenerate $ n $- dimensional spherically symmetric Gaussian vectors.

The idea behind the construction of an $ E $- space has been generalized. For example, if $ \Sigma $ is a set with some structure, e.g., a normed, inner product or topological space, then the set of all functions from $ ( \Omega, {\mathcal A}, {\mathsf P} ) $ into $ \Sigma $ 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 $ \Delta ^ {+} $. 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 $ f $ be a function from a metric space $ ( S,d ) $ into itself, and, for any non-negative integer $ m $, let $ f ^ {m} $ denote the $ m $ th iterate of $ f $. For any $ p,q \in S $, define the sequence $ \delta _ {pq } $ by

$$ \delta _ {pq } ( m ) = d ( f ^ {m} ( p ) ,f ^ {m} ( q ) ) , $$

and for any positive integer $ n $ define $ F _ {pq } ^ {( n ) } \in \Delta ^ {+} $ via

$$ F _ {pq } ^ {( n ) } ( t ) = $$

$$ = { \frac{1}{n} } \# \left \{ m : {0 \leq m \leq n - 1, \delta _ {m} ( p,q ) < t } \right \} , $$

where $ \# $ denotes the cardinality of the set in question. The number $ F _ {pq } ^ {( n ) } ( t ) $ may be interpreted as the probability that the distance between the initial segments, of length $ n $, of the trajectories of $ p $ and $ q $ is less than $ t $. Let

$$ F _ {pq } ( t ) = {\lim\limits \inf } _ {n \rightarrow \infty } F _ {pq } ^ {( n ) } ( t ) , $$

$$ F _ {pq } ^ {*} ( t ) = {\lim\limits \sup } _ {n \rightarrow \infty } F _ {pq } ^ {( n ) } ( t ) , $$

and let $ F _ {pq } $ and $ F _ {pq } ^ {*} $ be normalized to be left-continuous, hence in $ \Delta ^ {+} $. If $ {\mathcal F} $ is defined via $ {\mathcal F} ( p,q ) = F _ {pq } $, then, again, IV) holds with $ \tau = \tau _ {W} $. The resultant probabilistic pseudo-metric space is a transformation generated space. Note that $ F _ {pq } = F _ {f ( p ) f ( q ) } $, so that $ f $ is (probabilistic) distance-preserving.

If $ f $ is measure-preserving (cf. Measure-preserving transformation) with respect to a probability measure $ {\mathsf P} $ on $ S $, then $ F _ {pq } = F _ {pq } ^ {*} $ for almost all pairs $ ( p,q ) $ in $ S \times S $; and if, in addition, $ f $ is mixing, then there is a $ G \in \Delta ^ {+} $ such that $ F _ {pq } = F _ {pq } ^ {*} = G $ for almost all pairs $ ( p,q ) $.

The above ideas play an important role in chaos theory. For example, if $ S $ is a closed interval $ [ a,b ] $, if $ f $ is continuous, and if there is a single pair of points $ p,q \in [ a,b ] $ for which $ F _ {pq } \neq F _ {pq } ^ {*} $, then $ f $ is chaotic in a very strong sense. This fact leads to a theory of distributional chaos. Specifically, if $ ( S,d ) $ is compact, then $ f $ is distributionally chaotic if and only if there is a pair of points $ p,q \in S $ for which $ F _ {pq } \neq F _ {pq } ^ {*} $. Furthermore, the number

$$ \mu ( f ) = \sup _ {p,q \in S } { \frac{1}{d _ {S} } } \int\limits _ { 0 } ^ \infty {( F _ {pq } ^ {*} ( t ) - F _ {pq } ( t ) ) } {dt } , $$

where $ d _ {S} $ is the diameter of $ S $, 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)
How to Cite This Entry:
Probabilistic metric space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probabilistic_metric_space&oldid=26921
This article was adapted from an original article by B. SchweizerA. Sklar (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article