Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite measures on $\chi$, and $\mathcal{N}_{\chi}^{\#}$ refers to the space of boundedly finite counting measures on $\chi$.
A stationary random measure (respectively, point process) on state space $\chi=\mathbb{R}^d$ is said to be mixing if for all $V,W$ in $\mathcal{B}(\mathcal{M}_\chi^{\#})$ (respectively, $\mathcal{N}_\chi^{\#})$ we have $P(S_xV \cap W)-P(V)P(W) \to 0$, where $P$ is the probability measure of the random measure.
It is claimed in page 207 of this textbook (springer link) that any completely random measure such as the Poisson process clearly satisfies this mixing property.
Why is this claim true?