Poisson random measure

Let $$(E, \mathcal A, \mu)$$ be some measurable space with $$\sigma$$-finite measure $$\mu$$. The Poisson random measure with intensity measure $$\mu$$ is a family of random variables $$\{N_A\}_{A\in\mathcal{A}}$$ defined on some probability space $$(\Omega, \mathcal F, \mathrm{P})$$ such that

i) $$\forall A\in\mathcal{A}\;N_A$$ is a Poisson random variable with rate $$\mu(A)$$.

ii) If sets $$A_1,A_2,\ldots,A_n\in\mathcal{A}$$ don't intersect then the corresponding random variables from i) are mutually independent.

iii) $$\forall\omega\in\Omega\;N_{\bullet}(\omega)$$ is a measure on $$(E, \mathcal A)$$

Existence
If $$\mu\equiv 0$$ then $$N\equiv 0$$ satisfies the conditions i)-iii). Otherwise, in the case of finite measure $$\mu$$ given $$Z$$ - Poisson random variable with rate $$\mu(E)$$ and $$X_1, X_2,\ldots$$ - mutually independent random variables with distribution $$\frac{\mu}{\mu(E)}$$ define $$N_{\bullet}(\omega) = \sum\limits_{i=1}^{Z(\omega)} \delta_{X_i(\omega)}(\bullet)$$ where $$\delta_c(A)$$ is a degenerate measure located in $$c$$. Then $$N$$ will be a Poisson random measure. In the case $$\mu$$ is not finite the measure $$N$$ can be obtained from the measures constructed above on parts of $$E$$ where $$\mu$$ is finite.

Applications
This kind of random measures are often used when describing jumps of stochastic processes, in particular in Lévy-Itō decomposition of the Lévy processes.