Grand-canonical Gibbs measure for continuous systems

Question

Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+\infty\}$ is function which represents the potential energy of a system consisting in $N$ interacting particles. Now, the partition function of such a system in the grand-canonical ensemble is given by: \begin{eqnarray} \Xi_{\Lambda}(\beta, z) := \sum_{N=0}^{\infty}\frac{z^{N}}{N!} \int_{\Lambda}d\mu(x_{1})\cdots \int_{\Lambda}d\mu(x_{N}) e^{-\beta U_{N}(x_{1},...,x_{N})} \tag{1}\label{1} \end{eqnarray}

My Question is: What is the $\sigma$-algebra in which the product measures $\mu(x_{1})\times\cdots\times \mu(x_{N})$ are defined? I mean, for each $N$, the $N$-th term of the sum (\ref{1}) involves a product of $N$ measures and I'm having trouble understanding what is happening here. It seems to me that this is just a weak-limit in the sense that: \begin{eqnarray} \int d\mu_{n} f \to \int d\mu f \tag{2}\label{2} \end{eqnarray} but this implies that each $\mu_{n} = \mu(x_{1})\times \cdots \times \mu(x_{N}) $ is defined, for every $N$, on a "bigger" $\sigma$-algebra. What is this $\sigma$-algebra?

Answer 1

Looking at your formula (1), it appears that $\mu$ must be a measure defined on a $\sigma$-algebra $\mathscr F$ over the finite set $\Lambda$. The natural $\sigma$-algebra over the finite set $\Lambda$ is the largest $\sigma$-algebra over $\Lambda$, which is the (power) set $2^\Lambda$ of all subsets of $\Lambda$. By definition, the product measure $\mu^{\otimes N}$ is defined on the product $\sigma$-algebra $\mathscr F^{\otimes N}$. If $\mathscr F=2^\Lambda$, then $\mathscr F^{\otimes N}=(2^\Lambda)^{\otimes N}=2^{\Lambda^N}$, the set of all subsets of $\Lambda^N$.

Your formula (1) can then be rewritten simply as $$ \Xi_{\Lambda}(\beta, z) := \sum_{N=0}^{\infty}\frac{z^{N}}{N!} \int_{\Lambda^N}e^{-\beta U_{N}(x_{1},...,x_{N})}\,\mu^{\otimes N}\Big(\prod_{j=1}^N dx_j\Big) \\ =\sum_{N=0}^{\infty}\frac{z^{N}}{N!} \sum_{(x_{1},...,x_{N})\in\Lambda^N}e^{-\beta U_{N}(x_{1},...,x_{N})}\,\prod_{j=1}^N \mu(\{x_j\}). $$

Answer 2

The configuration space is the disjoint union of $\Lambda^N$ for each nonnegative integer $N$. You can take the Borel $\sigma$-algebra on each of these (or Lebesgue if you prefer, but you're unlikely to encounter non-Borel sets in real life).