Let $\mathbb{S}^1=\{x\in \mathbb{R}^2: \lvert x \rvert=1\}$. Consider $\mathcal{B}(\mathbb{S}^1)=\{B\cap\mathbb{S}^1:B \in \mathcal{B}(\mathbb{R}^2)\}$. I want to prove that there exists a measure on $\mu$$(\mathbb{S}^1,\mathcal{B}(\mathbb{S}^1))$, such that if $B\in \mathcal{B}(\mathbb{S}^1)$ is an arch, then $\mu(B)$ is its angle. So we must for example have $\mu(\mathbb{S}^1)=2\pi$. This statement is from my lecture notes.
What is the easiest way of prove this? I was thinking of defining the algebra of disjoint (closed, open?) archs and definiting a function to be the sum of its archs, then showing it is a premeasure and then using Carathéodory's extension theorem. I think it is not difficult to show that it is an algebra, but I don't know how to show that it is a premeasure.
Is this proved somewhere, could you please provide a link if so? Or is there a quicker way of getting the result than constructing the measure?