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Let $\Pi: ( \Omega, \mathcal{F}, \mathbb{P} ) \rightarrow \mathbb{R}^d$ be a Poisson point process.

We know that $\Pi_0=\{\left \| X \right \|, X\in \Pi\}$ is a Poisson point process on $\mathbb{R}_+$ and that $\phi(\Pi_0)$ ($\phi(r)=v_dr^d, v_d~~ \text{is the volume of the unit ball in}~~\mathbb{R}^d $) is a Poisson point process on $\mathbb{R}_+$ with intensity $l_1$ the Lebesgue measure on $\mathbb{R}_+$.

How can we show that there exists a sequence $(X_n)_{n\geq 1}$ of $\mathcal{F}$-measurable random variables such that $0< \left \| X \right \|_n< \left \| X \right \|_{n+1} ,\forall n\in\mathbb{N}^*$ and $ \Pi=\{X_n, n\in\mathbb{N}^* \} ~\mathbb{P}-\text{as}$. What is the Cumulative distribution function of $\left \| X_n \right \|$.

Can someone give a hint?

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