I want to construct a Haar measure on $\mathbb{C} \setminus 0$. That is, a Borel measure $\mu$ on $\mathbb{C} \setminus 0$ such that $\mu(zS) = \mu(S)$ for all $z \in \mathbb{C} \setminus 0$ and all Borel sets $S$.
I want to take the following approach. Identify $\mathbb{C} \setminus 0$ with the set $\mathbb{R}^+ \times \mathbb{T}$ where $\mathbb{R}^+ = \{x \in \mathbb{R} : x > 0\}$ and $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$ in the usual way using polar coordinates.
I know a Haar measure on $\mathbb{R}^+$.One can take $d\lambda(x) = dx/x$ where $dx$ is the Lebesgue measure.
I also know the standard Lebesgue measure restricted to $\mathbb{T}$ is invariant under multiplication by elements of $\mathbb{T}$.
How can I combine this data to construct a Haar measure on $\mathbb{C}$?