Random Finite Sets

Question

I am interested in a concept called ''random finite set'' (see Page 167 or Page 20).

Consider a probability space $(\Omega, \mathfrak{F}, \mathbb{P})$ and denote a subset of $\mathbb{R}^n$ by $\mathcal{X}$. Also, we denote the space of finite subsets of $\mathcal{X}$ by $\mathcal{F} (\mathcal{X})$. Suppose $\mathbb R^n$ is equipped with the a metric $d$ (it can be the usual metric). Then we can use Hausdorff metric to define a topology on $\mathcal{F} (\mathcal{X})$, denoted by $\tau_{H,d}$. (In the two articles I gave, they talked about myopic topology on $\mathcal{F} (\mathcal{X})$, which is actually the same as what we defined here.) After this, we can have use $\tau_{H, d}$ to generate Borel $\sigma$-algebra, denoted by $\mathfrak{B}_{H,d}$.

According to the general definition in the two articles I gave in the first paragraph, a random finite $X$, is a measurable mapping from $(\Omega, \mathfrak{F}, \mathbb{P})$ to $(\mathcal{F} (\mathcal{X}), \mathfrak{B}_{H,d})$.

Also, they gave an easier way to describe it. Since the cardinality of $X$ is finite, we can first use $\rho:\mathbb{N}\to \mathbb{R}$ to denote the discrete distribution of the cardinality of the random finite set. After ''fixing'' the cardinality, assumed to be $n$, we can use a probability distribution $P_n$ on the product space $\mathcal{X}^n = \mathcal{X}\times \cdots \times \mathcal{X}$ to describe the behavier of the random set.

Now, my question is: is the first general definition compatible with the second? Or how we connect these two concepts after assuming $\mathbb R^n$ equipped with 2-norm? Also, The structure of the borel $\sigma$-algbra $\mathfrak{B}_{H,d}$ is not very clear to me, since it depending on the choice of the metric $d$ is uncomfortable for me.

Any comments help.

Answer 1

First, let me assuage your concerns about $\mathfrak{B}_{H,d}$ potentially depending on the metric. As long as two metrics $d$ and $d'$ are compatible, then it is easy to see the induced Hausdorff metrics are equivalent, at least when restricted to $\mathcal{F} (\mathcal{X})$. To see this, observe that a sequence of sets $X_n$ converges to a finite set $X=\{x_1,\dots,x_k\}$ if any only if for every set of neighborhoods $U_i\ni x_i$, we have for sufficiently large $n$ that $X_n\subseteq \bigcup_{i=1}^k U_i$, and $X_n\cap U_i\neq \emptyset$ for each $i$. Therefore the topology of $\mathcal F(\mathcal X)$ (and thus also the Borel $\sigma$-algebra for $\mathcal F(\mathcal X)$) depends only on the original topology, not the choice of metric.

As for the compatibility of the definitions, everything should work fine as long as whichever original metric $d$ you put on $\mathbb R^n$ is separable, or at least makes $\mathcal X$ separable.

To translate back and forth between the definitions, first observe that a distribution in the sense of the first definition is a measure $\nu$ on $\mathcal{F}(\mathcal X)$. Now let $\mathcal F_k(\mathcal X)=\{X\in\mathcal F(\mathcal X)\mid \#X=k\}$, and observe that since the set of members of $\mathcal F(\mathcal X)$ with at most $k$ members is easily seen to be closed, the sets $\mathcal F_k(\mathcal X)$ are Borel, and so we may decompose $\nu$ as a sum of its restrictions $\nu_k=\nu\llcorner_{\mathcal F_k(\mathcal X)}$.

Now for each $k$, we may define $\rho(k)=\nu_k(\mathcal F(\mathcal X))=\nu(\mathcal F_k(\mathcal X))$. We also define $\mathcal X^k_0\subseteq \mathcal X^k$ to be the subset on which all coordinates are distinct. We then define a symmetric probability distribution $\mu_k$ on $\mathcal X^k_0$ by $$\mu_k(S)=\frac{\int \#(\pi_k^{-1}(t)\cap S)\,d\nu_k(t)}{\rho(k)k!}\text{.}$$ Here $\pi_k\colon \mathcal X^k_0\to \mathcal F_k(\mathcal X)$ is the projection. (You can address any concerns about the measurability of the integrand $t\mapsto\#(\pi_k^{-1}(t)\cap S)$ by observing that $\mathcal X$ is separable and $\pi_k$ is a local homeomorphism - indeed a $k!$-to-$1$ covering map.)

In the reverse direction, given symmetric probability distributions $\mu_k$ on $\mathcal X^k_0$ and a probability mass function $\rho\colon \mathbb N\to \mathbb R$, simply define $\nu_k=\rho(k)\pi_{k*}\mu_k$, and $\nu=\sum_k \nu_k$.

Remark

There is a slight subtlety in moving between definitions, namely that when moving from the first definition to the second, if $\nu_k=0$ then we define $\rho(k)=0$, but then our formula for $\mu_k$ doesn't make sense. This is no matter, since we can define $\mu_k$ however we like, or leave it undefined, as it will always be weighted by $0$ in any relevant calculations. This makes the second definition somewhat inelegant (to make it truly rigorous we have to talk about a family of not-necessarily defined measures), but still serviceable for any practical purposes.