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?