Example

Let us take $G = \mathbf{R}^n$ and for $\mu$ the Lebesgue measure; all of the hypotheses of the Cor. of Prop. 9 are satisfied. The discrete subgroups $H$ of $G$ such that $G/H$ is compact are none other than the discrete subgroups of rank $n$ (GT, VII, §1, No. 1, Th. 1); such a subgroup $H$ is generated by a basis $(a_i)_{1 \leq i \leq n}$ of $\mathbf{R}^n$, and

(the determinant being taken with respect to the canonical basis of $\mathbf{R}^n$) (Ch. VII, §2, No. 10, Th. 4). The space $D_c$ can be interpreted here in the following way: every subgroup $H \in D_c$ is the transform $g \cdot \mathbf{Z}^n$ of the subgroup $\mathbf{Z}^n$ by an element $g \in \mathbf{GL}(n, \mathbf{R})$, and the subgroup of $\mathbf{GL}(n, \mathbf{R})$ leaving $\mathbf{Z}^n$ stable may be identified with $\mathbf{GL}(n, \mathbf{Z})$. Consequently $D_c$ may be canonically identified, as a (non-topological) homogeneous space, with $\mathbf{GL}(n, \mathbf{R})/\mathbf{GL}(n, \mathbf{Z})$. On the other hand, $\mathbf{GL}(n, \mathbf{R})$ operates continuously in $\mathbf{R}^n$, hence also in $\mathscr{M}_+(\mathbf{R}^n)$ for the vague topology (§3, No. 3, Prop. 13), hence in the subspace $N_c$ of $\mathscr{M}_+(\mathbf{R}^n)$; moreover, the canonical homeomorphism (No. 4, Prop. 8) of $N_c$ onto $D_c$ is compatible with the laws of operation of $\mathbf{GL}(n, \mathbf{R})$. Since $\mathbf{GL}(n, \mathbf{R})$ is countable at infinity and $D_c$ is locally compact, the bijection of $\mathbf{GL}(n, \mathbf{R})/\mathbf{GL}(n, \mathbf{Z})$ onto $D_c$ defined above is a homeomorphism (Ch. VII, App. I, Lemma 2). The Cor. of Prop. 9 therefore gives a criterion for compactness in the homogeneous space $\mathbf{GL}(n, \mathbf{R})/\mathbf{GL}(n, \mathbf{Z})$.