I was reading a (brief) introduction about measure theory today and came across the following statement:
(Lebesgue measure on $\mathbb{R}/\mathbb{Z}$): There is a unique probability measure $\mu$ on $\mathbb{R}/\mathbb{Z}$ such that $\mu((a,b))=b-a$ for all $0\le a.
My question is not about the proof (the notes are too brief to have the proofs) but about the $\sigma$-algebra in question. I assume that $(\mathbb{R}/\mathbb{Z},B,\mu)$ is a measure space where $B$ is the Borel $\sigma$-algebra, i.e. the smallest $\sigma$-algebra containing all the open balls. What is the metric in question? How come all elements of $B$ of the form $(a,b)$? Do we use the bijection between $\mathbb{R}/\mathbb{Z}$ and $[0,1)$ somewhere?
I'll be grateful if someone can clarify my doubts. Thanks.