$\mathcal{S}^{1}:=\{ (x,y)\in \mathbb R^{2}: x^{2}+y^{2}=1\}$
Show that the rotation invariant measure on $\mathcal{S}^{1}$ has uniform distribution. Via Polar Coordinates $(\theta \mapsto (\cos{(\theta)},\sin{(\theta)}))$ I want to consider the generator $\mathcal{C}:=\{[0,\beta[: \beta \in ]0,2\pi[\}$ of $\mathcal{B}(\mathcal{S}^{1})$
Then by definition of $P$ being a rotation-invariant measure, we have
$P( [0,\beta[)=P([a,a+\beta[)$ for any $a \in \mathbb R$. By this definition, it is clear the measure has to be uniformly distributed, I am just unsure on how to substantiate it.
Assume $\ P\ $ is normalised so that $\ P\left(\mathcal{S}^1\right)=1\ $. Then \begin{align} P\left(\mathcal{S}^1\right)&=\sum_{k=0}^{n-1}P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\frac{2\pi k}{n}, \frac{2\pi (k+1)}{n}\right[\right.\right\}\right)\\ &=nP\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[0, \frac{2\pi}{n}\right[\right.\right\}\right)\ , \end{align} by the invariance of $\ P\ $. Therefore $$ P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\alpha,\alpha + \frac{2\pi}{n}\right[\right.\right\}\right)=\frac{1}{n}\ . $$ Now, for $\ m\le n\ $, $$ \hspace{-8em}P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[0, \frac{2\pi m}{n}\right[\right.\right\}\right)=\\ \hspace{4em}\sum_{k=0}^{m-1}P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\frac{2\pi k}{n}, \frac{2\pi (k+1)}{n}\right[\right.\right\}\right)\\ = \frac{m}{n}\ , $$ again, by the invariance of $\ P\ $. Thus, $$ P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\alpha,\alpha + 2\pi q\right[\right.\right\}\right)=q\ , $$ for all rational $\ q\in[0,1]\ $.
Now if $\ 0<\beta\le\alpha+2\pi\ $ then there is an increasing sequence of rational numbers $\ \left\{r_k\right\}_{k=1}^\infty\ $ with $\ r_1>\alpha\ $ and $\ \lim_\limits{k\rightarrow\infty}r_i=\frac{\beta-\alpha}{2\pi}\ $, so \begin{align} P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\alpha,\alpha + \beta \right[\right.\right\}\right)&= P\left(\bigcup_\limits{k=1}^\infty\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\alpha,\alpha + 2\pi r_k\right[\right.\right\}\right)\\ &=\lim_\limits{k\rightarrow\infty}P\left(\left\{\,(\cos\theta,\sin\theta)\left\vert \,\theta\in\left[\alpha,\alpha + 2\pi r_k\right[\right.\right\}\right)\\ &=\frac{\beta-\alpha}{2\pi}\ , \end{align} which establishes the uniformity of $\ P\ $.