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.
