# Haar Measure on topological groups

I'm currently reading an article and the author defines the following objects. Let $$\mathbb{Z}_{n}$$ be the cyclic group of integers mod $$n$$, for some $$n \ge 1$$ and define $$\mathcal{G} := \bigoplus_{k=0}^{\infty}\mathbb{Z}_{n}$$ Furthermore, define the subgroups: $$\mathcal{G}_{k} :=\{x \in \mathcal{G}, \hspace{0.1cm} \mbox{x_{i}=0, for all i \ge k}\}$$ if $$k\ge 1$$ and $$\mathcal{G}_{0}:=\{0\}$$. Now, if $$L\ge 2$$, we introduce a notion of norm: $$|x| :=\begin{cases} \displaystyle 0 \quad \mbox{if x=0}\\ \displaystyle L^{p} \quad \mbox{where p=\inf \{k,\hspace{0.1cm} x\in \mathcal{G}_{k}\} if x \neq 0} \end{cases}$$ Now, the author defines, in his words "$$dx$$ to be a Haar measure which is also the counting measure on $$\mathcal{G}$$". I'm not familiar with the concept of a Haar measure on a group, but I've read that it should be defined on some locally compact Hausdorff topological group. Now, how do I conclude that $$\mathcal{G}$$ is such a group? Does $$|\cdot|$$ define a topology on $$\mathcal{G}$$? How can I define a topology on $$\mathcal{G}$$ with the above informations? I imagine $$|\cdot|$$ defines a topology in some way, but I don't know how. Besides, does this topology fulfill all properties (Hausdorff, locally compact) to define the Haar measure?

• The Haar measure is the unique (up to a multiple) measure invariant under left translations. On $\mathbb{Z}_k$ the counting measure is clearly invariant under addition. So you can take the “direct sum” of those measures for $\mathcal{G}$. – mysatellite Nov 25 '19 at 20:03
• The topology on your group is just the discrete topology (and the norm is also discrete). – YCor Nov 25 '19 at 21:38
• Means I have to take the set of all subsets of $\mathcal{G}$ as my $\sigma$-algebra? – IamWill Nov 25 '19 at 21:40