# Measure, volume and cardinality on Minlos' book on statistical physics

The following content was based on Minlos' book on statistical physics. Let $$\Lambda \subset \mathbb{R}^{d}$$ be fixed (Minlos takes $$d=3$$ but I think the ideas follow without change to $$d \ge 1$$). We denote by $$\Lambda^{N}$$ the $$N$$-fold cartesian product of $$\Lambda$$ with itself and $$(\Lambda^{N})'$$ $$N$$-uples $$(x_{1},...,x_{N})$$ in $$\Lambda$$ with different entries, i. e. $$x_{i}\neq x_{j}$$ if $$i\neq j$$. Also $$\Gamma_{\Lambda, N}:=\{\omega \subset \Lambda, \hspace{0.1cm} \mbox{card}(\omega) = N\}$$, where $$\mbox{card}(\omega)$$ is the cardinality of the set $$\omega$$. Define $$\Pi: (\Lambda^{N})' \to \Gamma_{\Lambda, N}$$ by $$(x_{1},...,x_{N}) \mapsto \{x_{1},...,x_{N}\}$$. For every subset $$A$$ of $$\Gamma_{\Lambda, N}$$, Minlos set: $$\mu_{\Lambda}^{(N)}(A) := \frac{\mbox{Vol}(\Pi^{-1}(A))}{N!}$$ Then, he states that $$\mu_{\Lambda}(\Gamma_{\Lambda, N}) = \frac{|\Lambda|^{N}}{N!}$$. The problem is that he doesn't seem to define $$\mbox{Vol}$$ or $$|\cdot|$$ anywhere and it is getting me a little confused. At first, I thought $$\mbox{Vol}$$ was just the Lebesgue measure on $$\mathbb{R}^{dN}$$. But it would be a little odd because if I take $$A = \{x_{1},...,x_{N}\}$$ it seems that $$\mu_{\Lambda}^{(N)}(\{x_{1},...,x_{N}\}) = 0$$. Besides, how come does the second statement about $$\mu_{\Lambda}(\Gamma_{\Lambda, N})$$ follow? If $$|\Lambda|$$ is the cardinality of $$\Lambda$$ (which I don't know for sure), does it follow from de definition of $$\mu_{\Lambda}$$?

If $$\text{Vol}$$ denotes the Lebesgue measure on $$(\mathbb R^d)^N$$, if $$|\cdot|$$ denotes the Lebesgue measure on $$\mathbb R^d$$, and if $$\mu_\Lambda=\mu_\Lambda^{(N)}$$, then indeed $$\mu_\Lambda(\Gamma_{\Lambda,N})=\frac{|\Lambda|^N}{N!}$$. This follows because (i) $$\Pi^{-1}(\Gamma_{\Lambda,N})=(\Lambda^N)'$$ and (ii) (say, by the Fubini--Tonelli theorem) $$\text{Vol}((\Lambda^N)')=\text{Vol}(\Lambda^N)=|\Lambda|^N$$.
As for $$\mu_\Lambda^{(N)}(A)$$ with $$A=\{x_{1},...,x_{N}\}$$, it is undefined. Indeed, for $$\mu_\Lambda^{(N)}(A)$$ be defined, $$A$$ must be a subset of $$\Gamma_{\Lambda,N}$$, whereas $$\{x_{1},...,x_{N}\}$$ is a subset of $$\mathbb R^d$$ but not of $$\Gamma_{\Lambda,N}$$. On the other hand, if $$A$$ is the singleton set $$\{(x_{1},...,x_{N})\}$$ with pairwise distinct $$x_j$$'s in $$\mathbb R^d$$, then indeed $$\mu_\Lambda^{(N)}(A)=0$$, and there is nothing odd about that.
• This is a very good answer! In summary, it looks like $|\cdot|$ is not the cardinality of the set, but it $|\cdot|$ and $\mbox{Vol}$ seem to be two Lebesgue measures on different spaces. It makes sense to me! – MathMath Feb 9 at 13:52