Exercise:Assume that $\Omega$ is a circumference of radius $1$ and centred at the origin of $\mathbb{R}^2$. Show that there exists a unique measure $\mu$ defined on $\mathscr{B}_{\Omega}$ such that $\mu(\Omega)=1$ and $\mu$ is invariant for all rotations centred at the origin.
I tried to solve the question the following way: I can define de measure using the Lebesgue measure $\mu(A)=\frac{\lambda(A)}{\lambda(\Omega)}=\frac{\lambda(A)}{2\pi}$ for $A \subseteq\Omega$
If $A_j$ is a disjoint sequence of sets so that $\bigcup_{j\in\mathbb{N}}A_j=\Omega$
If there were two measures $\mu_1$ and $\mu_2$ then since by assumption $\mu_1(\Omega)=1$ and $\mu_2(\Omega)=1$ then $\mu_2(\bigcup_{j\in\mathbb{N}}A_j)=\sum_{j\in\mathbb{N}}\mu_2(A_j)=\mu_2(\Omega)=1=\mu_1(\Omega)=\mu_1(\bigcup_{j\in\mathbb{N}}A_j)=\sum_{j\in\mathbb{N}}\mu_1(A_j)$
Since the measure is invariant to the rotations I think it does not matter the measure attributed to each single $A_j$, so what it needs to be assured in order to have uniqueness is that both $\mu_1$ and $\mu_2$ attribute the same set of values to the different sets. Otherwise if one was the Dirac measure and the other was not the uniqueness would fail.
Question:
How should I solve the problem?
Thanks in advance!