Let
(
M
,
d
)
{\displaystyle (M,d)}
complete separable metric space . Let
K
{\displaystyle {\mathcal {K}}}
M
{\displaystyle M}
h
{\displaystyle h}
K
{\displaystyle {\mathcal {K}}}
h
(
K
1
,
K
2
)
:=
max
{
sup
a
∈
K
1
inf
b
∈
K
2
d
(
a
,
b
)
,
sup
b
∈
K
2
inf
a
∈
K
1
d
(
a
,
b
)
}
.
{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}
(
K
,
h
)
{\displaystyle ({\mathcal {K}},h)}
σ-algebra on
K
{\displaystyle {\mathcal {K}}}
Borel sigma algebra
B
(
K
)
{\displaystyle {\mathcal {B}}({\mathcal {K}})}
K
{\displaystyle {\mathcal {K}}}
A random compact set is а measurable function
K
{\displaystyle K}
probability space
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
(
K
,
B
(
K
)
)
{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}
Put another way, a random compact set is a measurable function
K
:
Ω
→
2
M
{\displaystyle K\colon \Omega \to 2^{M}}
K
(
ω
)
{\displaystyle K(\omega )}
almost surely compact and
ω
↦
inf
b
∈
K
(
ω
)
d
(
x
,
b
)
{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}
is a measurable function for every
x
∈
M
{\displaystyle x\in M}
Random compact sets in this sense are also random closed sets as in Matheron (1975). Consequently, under the additional assumption that the carrier space is locally compact, their distribution is given by the probabilities
P
(
X
∩
K
=
∅
)
{\displaystyle \mathbb {P} (X\cap K=\emptyset )}
K
∈
K
.
{\displaystyle K\in {\mathcal {K}}.}
(The distribution of а random compact convex set is also given by the system of all inclusion probabilities
P
(
X
⊂
K
)
.
{\displaystyle \mathbb {P} (X\subset K).}
For
K
=
{
x
}
{\displaystyle K=\{x\}}
P
(
x
∈
X
)
{\displaystyle \mathbb {P} (x\in X)}
P
(
x
∈
X
)
=
1
−
P
(
x
∉
X
)
.
{\displaystyle \mathbb {P} (x\in X)=1-\mathbb {P} (x\not \in X).}
Thus the covering function
p
X
{\displaystyle p_{X}}
p
X
(
x
)
=
P
(
x
∈
X
)
{\displaystyle p_{X}(x)=\mathbb {P} (x\in X)}
x
∈
M
.
{\displaystyle x\in M.}
Of course,
p
X
{\displaystyle p_{X}}
1
X
{\displaystyle \mathbf {1} _{X}}
p
X
(
x
)
=
E
1
X
(
x
)
.
{\displaystyle p_{X}(x)=\mathbb {E} \mathbf {1} _{X}(x).}
The covering function takes values between
0
{\displaystyle 0}
1
{\displaystyle 1}
b
X
{\displaystyle b_{X}}
x
∈
M
{\displaystyle x\in M}
p
X
(
x
)
>
0
{\displaystyle p_{X}(x)>0}
support of
X
{\displaystyle X}
k
X
{\displaystyle k_{X}}
x
∈
M
{\displaystyle x\in M}
p
X
(
x
)
=
1
{\displaystyle p_{X}(x)=1}
kernel , the set of fixed points , or essential minimum
e
(
X
)
{\displaystyle e(X)}
X
1
,
X
2
,
…
{\displaystyle X_{1},X_{2},\ldots }
i.i.d. random compact sets, then almost surely
⋂
i
=
1
∞
X
i
=
e
(
X
)
{\displaystyle \bigcap _{i=1}^{\infty }X_{i}=e(X)}
and
⋂
i
=
1
∞
X
i
{\displaystyle \bigcap _{i=1}^{\infty }X_{i}}
e
(
X
)
.
{\displaystyle e(X).}
Matheron, G. (1975) Random Sets and Integral Geometry . J.Wiley & Sons, New York. ISBN 0-471-57621-2
Molchanov, I. (2005) The Theory of Random Sets . Springer, New York. ISBN 1-85233-892-X
Stoyan D., and H.Stoyan (1994) Fractals, Random Shapes and Point Fields . John Wiley & Sons, Chichester, New York. ISBN 0-471-93757-6
