I am having trouble in understanding this definition of Poisson process.
Let $S$ be a random discrete subset of points of $\mathbb{R}^d$ and let $\lambda >0$.
A partition $\mathcal{A}$ of $\mathbb{R}^d$ with $A\in \mathcal{A}$ measurable and $l(A)<\infty$.
Independent Poisson random variables $Y_A\sim\text{Poisson}(\lambda l(A))$.
A family $((U_{A,j}, j\ge 1) A\in \mathcal{A})$, where $U_{A,j}\sim\text{Unif}(A)$ independent.
Define $S=\bigcup_{A\in \mathcal{A}}\bigcup_{j\le Y_A}\{U_{A,j}\}$
$S$ is a Poisson process of intensity $\lambda$.
All I already knew was the definition given in the wikipedia page
Are these two different or have connection? Can some one help understanding this?