Poisson random measure
Let ( E , A , μ ) {\displaystyle (E,{\mathcal {A}},\mu )} be some measure space with σ {\displaystyle \sigma } -finite measure μ {\displaystyle \mu } . The Poisson random measure with intensity measure μ {\displaystyle \mu } is a family of random variables { N A } A ∈ A {\displaystyle \{N_{A}\}_{A\in {\mathcal {A}}}} defined on some probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathrm {P} )} such that i) ∀ A ∈ A , N A {\displaystyle \forall A\in {\mathcal {A}},\quad N_{A}} is a Poisson random variable with rate μ ( A ) {\displaystyle \mu (A)} .