I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is some little difference between his and Minlo's book that I'd like to clarify.

On page 27, Adams introduces two sets, as follows (with $\Lambda \subset \mathbb{R}^{d})$: $$\Gamma_{\Lambda, N} :=\{\omega \subset \Lambda \times \mathbb{R}^{d}, \omega =\{(q,p_{q}), q \in \hat{\omega}\}, |\hat{\omega}|=N\} \quad (1)$$ $$ \Gamma_{\Lambda} := \{\omega \subset \Lambda \times \mathbb{R}^{d}, \omega = \{(q,p_{q}), q \in \hat{\omega}\}, |\hat{\omega}|<+\infty\} \quad (2) $$ In addition, for each $\Delta \subset \Lambda$ Borel-measurable, define a counting variable $N_{\Delta}$ to be the function $N_{\Delta}:\Gamma_{\Lambda} \to \mathbb{R}$ by $N_{\Delta}(\omega) = |\omega \cap \Delta|$. The $\sigma$-algebra on $\Gamma_{\Lambda}$ generated by the family of counting variables $\{N_{\Delta}\}$ is denoted by $\mathcal{B}_{\Lambda}^{\infty}$.

Then, the notes procceeds by introducing the following definition.

**Definition:** Let $\Lambda \subset \mathbb{R}^{d}$, $\beta > 0$ and $\mu \in \mathbb{R}$. Define the phase space $\Gamma_{\Lambda}:=\bigcup_{N=0}^{\infty}\Gamma_{\Lambda, N}$, where $\Gamma_{\Lambda, N} := (\Lambda\times \mathbb{R}^{d})^{2N}$ is the phase space of exactly $N$ particles and equip it with $\mathcal{B}_{\Lambda}^{\infty}$. The probability mesure $\gamma_{\Lambda, \beta}$ on $(\Gamma_{\Lambda},\mathcal{B}_{\Lambda}^{\infty})$ such that the restrictions $\gamma_{\Lambda, \beta}|_{\Gamma_{\Lambda, N}}$ have densities:
$$\rho_{\Lambda,\beta}^{(N)}(x) = Z_{\Lambda}(\beta, \mu)^{-1}e^{-\beta H_{\Lambda}^{(N)}(x)-\mu N}$$
where $H_{N}^{(N)}$ is the Hamiltonian for $N$ particles in $\Lambda$ is called grand canonical ensemble in $\Lambda$.

Well, Adams seems to be using two different notions of phase space at the same time. In the above definition, $\Gamma_{\Lambda} = \bigcup_{N=0}^{\infty}\Gamma_{\Lambda,N}$ but the $\sigma$-algebra $\mathcal{B}_{\Lambda}^{\infty}$ only makes sense in $\Gamma_{\Lambda}$ given by (2). Thus, it seems to me that he's using some identification between $\Gamma_{\Lambda,N}$ as given by (1) and $(\Lambda \times \mathbb{R}^{d})^{2N}$. But what is this idetification? Note that nothing prevents us to take an element in $(\Lambda \times \mathbb{R}^{d})^{2N}$ which has equal entries and this would led to a single point in $\Gamma_{\Lambda,N}$ as given by (1). What m I getting wrong here?