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Reading a reply by Didier, I was wondering

  1. How is a "random subset" defined? Is it a measurable mapping, i.e., random element, from a probability space to the power set of another set, with respect to some sigma algebra(?) defined on the power set?
  2. How is a random subset being locally finite defined?

Thanks and regards!

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    @Rasmus: are you talking about random subset?2011-04-20

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A random subset $\mathcal{N}$ being (almost surely) locally finite means that $\#(\mathcal{N}\cap B)$ is (almost surely) finite for every compact subset $B$ of the target space. In the setting of point processes, one considers only (almost surely) locally finite random subsets $\mathcal{N}$ so, in a way, one avoids at all cost the power set of the target space, which is much too big for measurability purposes.

The distribution of a locally finite random subset $\mathcal{N}$ is defined by the distributions of the finite families of integer valued random variables $\#(\mathcal{N}\cap B)$ for compact subsets $B$, just like the distribution of an infinite sequence $(\xi_n)_n$ indexed by the integers is defined by the distributions of the random vectors $(\xi_n)_{n\in I}$ for every finite $I$, aka the marginals of the process. Note in particular that one assumes that $\#(\mathcal{N}\cap B)$ is measurable for every compact $B$.

When $\#(\mathcal{N}\cap B)$ is almost surely finite for every compact $B$, one can identify the random subset $\mathcal{N}$ with the random measure which puts a unit Dirac mass on every point in $\mathcal{N}$, defined formally bas the unique measure $N$ such that, at least for every measurable bounded function $f$ with bounded support, $ \int f(x)\, \mathrm{d}N(x)=\sum_{x\in\mathcal{N}}f(x). $ This identification goes through the (trivial) remark that, for every measurable subset $B$, the events $[\mathcal{N}\cap B=\emptyset]$ and $[N(B)=0]$ coincide.

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    Replace *bounded* by *compact*.2011-04-21