Three questions about Haar measure

Question

I have been reading on Haar measure recently.

Let $G$ be a locally compact group with Haar measure $\mu$.

1. $\mu(\{e\})>0$ then $G$ is discrete.
2. $\mu(G)<\infty$ then $G$ is compact.
3. we know that every locally compact Hausdorff group admits a Haar measure, is the same true for monoids(semigroup with identity e)? If not, is there any counterexample? Is there a class of semigroups that admits a Haar measure?

I just saw the "Finite Haar Measure if and only if Compact" by Gils.

About the first part, I can't understand what Gils said, If $\mu(\{e\})>0$ then $\mu$ is a scalar multiple of the counting measure. Since $\mu$ is outer-regular, this means that $\{e\}$ is open.

About the second part, I'd like a proof without any integrals.

About the third part, My question on Haar Measure on Locally Compact monoids hasn't been answered yet. You can answer this question on that page. I am willing to accept the answer.

Any help will be appreciated.

Answer 1

1. Suppose $\alpha := \mu(\{e\}) > 0$, then $\mu(\{x\}) = \alpha$ for all $x\in G$. Now if $\epsilon := \alpha/2 > 0, \exists U$ open such that $e\in U$ and $$ \mu(U) < \mu(\{e\}) + \epsilon $$ Conclude that $U = \{e\}$ must hold.

2. Suppose $G$ is not compact, then by local compactness, choose a neighbourhood $U$ of $e$ such that $\overline{U}$ is compact. Clearly, $\exists g_1 \in G\setminus U$. Now, $$ U\cup g_1U $$ has compact closure, so $\exists g_2 \in G\setminus (U\cup g_1U)$. Thus proceeding, we obtain a sequence $(g_n)$ such that $$ g_n \in G\setminus \left( \bigcup_{i=0}^{n-1} g_i(U)\right) $$ where $g_0 = e$. Now choose an open set $V$ such that $e\in V$ and $VV^{-1} \subset U$, then the sets $$ \{g_n V: n\in \mathbb{N}\} $$ are mutually disjoint. Since $V$ is open, $\mu(V) > 0$ and this would contradict the fact that $\mu(G) < \infty$.

3. See this :

https://mathoverflow.net/questions/180019/haar-measure-on-locally-compact-semigroups