# Random Finite Sets

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.

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.

• Thank you for your patient explanation! However, I am still a little confused about what is the topology ( or Borel algebra generated by this topology) of the second definition. Is it the same as that in the first definition by Hausdorff distance? I cannot see this very clearly. Anyway, thanks again! Sep 24 at 6:44
• @Greenhand In the second definition they talked about a family of distributions and a single probability mass function, so the second definition doesn't seem to be referring to a single distribution, therefore there isn't only a single $\sigma$-algebra at play, as I understand it. Instead you have a different Borel $\sigma$-algebras for each $k$, namely those corresponding to the Borel subsets of $\mathcal X_0^k$, and of course $\rho$ naturally determines a measure on the Borel $\sigma$-algebra for $\mathbb N$ with the discrete topology, which is just the power set $\mathcal P(\mathbb N)$.
– M W
Sep 24 at 7:00
• @M W You got the point. That is where I am always confused about, since the second one is not a typical case of the first general definition. Thus, I do not know why they introduced the myopic topology which they did not use. Sep 24 at 7:20