I am interested in the space $$ X:=\{x \subset \mathbb{R}^3: |x| < \infty\}, $$ where $|x|$ is the cardinality of the subset $x$. This is basically configuration space for a quantum system with a variable number of indistinguishable particle. In particular, I would like to define a metric (derived from the usual metric in $\mathbb{R}^n$), to know if the space is complete with respect to this metric, and to define a measure (derived from the usual Lebesgue measure in $\mathbb{R}^n$).
Answer to Daniel Rust. My basic idea is the following: define $\Gamma:=\bigcup_{n=0}^\infty \mathbb{R}^{3n}$, and then define $h:\Gamma \to X$ as follows: $h({\bf y}_1, \ldots, {\bf y}_n):=\{{\bf y}_1\} \cup \ldots \cup \{{\bf y}_n\}$. After this, for $y, y' \in \Gamma$, let $y \sim y'$ if $h(y)=h(y')$. There is an obvious bijection between $X$ and $\Gamma/\sim$, so one can define a suitable metric and a measure on $\Gamma/\sim$, based on the Euclidean metric and the Lebesgue measure.