Verifying Haar Measure

Question

I am reading Haar Measure right now.
Definition: For G be locally compact Hausdorff group. A left Haar functional on G is a non trivial positive linear functional on $C_{c}(G)$ which is invariant under left translation.
I am reading an example of G=$\mathbb{R^{*}}$. The map $D$ from $C_{c}(G)$ to $\mathbb{R}$ defined by $f \mapsto \int_\mathbb{R} f(x)\frac{dx}{|x|}$ is left and right Haar functional. It can be verified using classical substitution principle. In last it concluded that D defines a left and right Haar measure on G. I don’t know how this concluded. I have read a theorem that for every Haar functional on G we have a Haar measure in G and conversely. Having some hint or suggestion would really help.

Answer 1

Note that for any Borel set $A \subseteq \mathbb{R}^*$ we can define $$\mu (A) = \int_A \frac{1}{|x|} dx$$ where $dx$ stands for integration against the Lebesgue measure. This measure is Borel regular and locally finite; as advertised, you can argue by change of variables to show that this measure is translation invariant.

You may conclude that this is indeed a Haar measure. Notice that since $\mathbb{R}^*$ is an Abelian group this measure is, at once, the left and right Haar measure for this group.