Let
(
E
,
A
,
μ
)
be some measure space with
σ
-finite measure
μ
. The Poisson random measure with intensity measure
μ
is a family of random variables
{
N
A
}
A
∈
A
defined on some probability space
(
Ω
,
F
,
P
)
such that
i)
∀
A
∈
A
,
N
A
is a Poisson random variable with rate
μ
(
A
)
.
ii) If sets
A
1
,
A
2
,
…
,
A
n
∈
A
don't intersect then the corresponding random variables from i) are mutually independent.
iii)
∀
ω
∈
Ω
N
∙
(
ω
)
is a measure on
(
E
,
A
)
If
μ
≡
0
then
N
≡
0
satisfies the conditions i)–iii). Otherwise, in the case of finite measure
μ
, given
Z
, a Poisson random variable with rate
μ
(
E
)
, and
X
1
,
X
2
,
…
, mutually independent random variables with distribution
μ
μ
(
E
)
, define
N
⋅
(
ω
)
=
∑
i
=
1
Z
(
ω
)
δ
X
i
(
ω
)
(
⋅
)
where
δ
c
(
A
)
is a degenerate measure located in
c
. Then
N
will be a Poisson random measure. In the case
μ
is not finite the measure
N
can be obtained from the measures constructed above on parts of
E
where
μ
is finite.
This kind of random measure is often used when describing jumps of stochastic processes, in particular in Lévy–Itō decomposition of the Lévy processes.
