probabilistic metric space

probabilistic metric space

Recall that a

metric space

is a set

X

equipped with a

distance function

d

:

X

×

X

→

[

0

,

∞

)

, such that

1.

d

⁢

(

a

,

b

)

=

0

iff

a

=

b

,

2.

d

⁢

(

a

,

b

)

=

d

⁢

(

b

,

a

)

, and

3.

d

⁢

(

a

,

c

)

≤

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

Δ

+

. For example, for any

r

≥

0

, the

step functions

defined by

e

r

⁢

(

x

)

=

{

0

when

⁢

x

≤

r

,

1

otherwise

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

Δ

+

by

F

≤

G

iff

F

⁢

(

x

)

≤

G

⁢

(

x

)

for all

x

∈

ℝ

. It is not hard to see that

e

x

≤

e

y

iff

y

≤

x

and that

e

0

is the top element of

Δ

+

. From the poset

Δ

+

, call a binary operator

τ

on

Δ

+

a

triangle function

if

τ

turns

Δ

+

into a partially ordered (

http://planetmath.org/PartiallyOrderedGroup

)

commutative monoid

with

e

0

serving as the

identity element

. Spelling this out, for any

F

,

G

,

H

∈

Δ

+

, we have

•

F

⁢

τ

⁢

G

=

G

⁢

τ

⁢

F

,

•

(

F

⁢

τ

⁢

G

)

⁢

τ

⁢

H

=

F

⁢

τ

⁢

(

G

⁢

τ

⁢

H

)

,

•

F

⁢

τ

⁢

e

0

=

e

0

⁢

τ

⁢

F

=

F

, and

•

if

G

≤

H

, then

F

⁢

τ

⁢

G

≤

F

⁢

τ

⁢

H

,

where

F

⁢

τ

⁢

G

means

τ

⁢

(

F

,

G

)

. For example,

F

⁢

τ

⁢

G

=

F

⋅

G

,

F

⁢

τ

⁢

G

=

min

⁡

(

F

,

G

)

are two triangle functions. In fact, since

F

⁢

τ

⁢

G

≤

F

⁢

τ

⁢

e

0

=

F

and

F

⁢

τ

⁢

G

≤

G

similarly, we have

F

⁢

τ

⁢

G

≤

min

⁡

(

F

,

G

)

for any triangle function

τ

.

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

×

X

→

Δ

+

, where

Δ

+

is the set of distance distribution functions on which a triangle function

τ

is defined, such that

1.

F

(

a

,

b

)

=

e

0

iff

a

=

b

, where

F

(

a

,

b

)

:=

F

⁢

(

a

,

b

)

,

2.

F

(

a

,

b

)

=

F

(

b

,

a

)

, and

3.

F

(

a

,

c

)

≥

F

(

a

,

b

)

⁢

τ

⁢

F

(

b

,

c

)

.

Given a metric space

(

X

,

d

)

, if we can find a triangle function

τ

such that

e

x

⁢

τ

⁢

e

y

=

e

x

+

y

, then

(

X

,

F

)

with

F

(

a

,

b

)

:=

e

d

⁢

(

a

,

b

)

is a probabilistic metric space.

References

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. Učen. Zap. 122, 3-20, (1962).

Title

probabilistic metric space

Canonical

name

ProbabilisticMetricSpace

Date of creation

2013-03-22 16:49:38

Last modified on

2013-03-22 16:49:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E70

Defines

distance distribution function

Defines

triangle function

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