Let $G$ be a locally profinite group. A Haar measure $\mu$ on $G$ is a measure defined on the $\sigma$-algebra $\mathcal B(G)$ of all Borel sets of $G$ with the following properties.
1) $\mu(K) \lt \infty$ for every compact set $K$.
2) $\mu(U) = \text{sup} \{\mu(K) : K \subset U$, where $K$ is compact$\}$ for every open set $U$.
3) $\mu(E) = \text{inf} \{\mu(U): E \subset U$, where $U$ is open $\}$ for every $E\in \mathcal B(G)$.
4) $\mu(aE) = \mu(E)$ whenever $a\in G$ and $E\in \mathcal B(G)$.
My question is:
Can we explicitly construct a Haar measure on $G$?
I am almost certain that we can explicitly construct a Haar mesure on $G$ if $G$ is compact, i.e. if $G$ is a profinite group. I think the following threads are relevant to this problem.
Explicit construction of Haar measure on a profinite group
On formula $\sum_{i=1}^n 1/(G : H_i) = 1$ on a group $G$
Construction of a Radon measure from a certain family of compact subsets
