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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 $0\le a<b\le 1$.

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.

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It is not true that every element of $B$ is an open interval. Rather, the open intervals generate $B$ as a $\sigma$-algebra. –  Zhen Lin Jul 10 '12 at 11:10
    
Why is any element of $B$ an open interval? Aren't elements of $B$ subsets of $\{x+\mathbb{Z}:x\in\mathbb{R}\}$? –  Shahab Jul 10 '12 at 11:15
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@Shahab: There's a canonical injection $[0,1)\to \mathbb R/\mathbb Z$. The image of an open interval under this bijection is an element of $B$. Alternatively, you can consider $\mathbb R/\mathbb Z$ to be $[0,1)$ -- albeit with a nonstandard topology -- simply by always choosing the unique representative for each class that lies in this interval. –  Henning Makholm Jul 10 '12 at 11:25

1 Answer 1

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Yes, one uses the bijection you mention. To be precise, let $f:[0,1]\to S^1$ be the defining quotient map $t\mapsto e^{2\pi i t}$. Write $\lambda$ for Lebesgue measure on $[0,1]$.

Then the property is $\mu(S)=\lambda(f^{-1}(S))$, or $\mu(f(a,b))=\lambda(a,b):=b-a$. This is usually called Haar measure (rather than Lebesgue measure) on $S^1$.

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I believe when you call it "Haar measure" you look at it as the measure that is induced by the locally compact group structure, but when you call it "Lebesgue measure", you look at is as the measure that arises induced by the quotient map from the usual Lebesgue measure (or as curve measure in the plane). So the two names refer to different aspects of the measure. –  tomasz Jul 10 '12 at 13:47

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