I have a question regarding the proof of unique ergodicity of irrational rotations on the last page of this script (example §20.3)
http://www.maths.manchester.ac.uk/~cwalkden/ergodic-theory/lecture20.pdf
A bit of context first: The author wants to prove that both measures $\mu$ and $\nu$ are equal. He uses the result that two measures on the Borel-$\sigma$-algebra of a metric space are equal iff they integrate continuous functions in the same way. Therefore he considers a continuous function $f$ and writes it as the uniform limit of the averages $\sigma_n$ of the first $n$ partial sums of the Fourier series. It is not stated, but I assume that he's making use of Fejér's theorem.
Now here is my problem: I know that $C(\mathbb{T})$ is usually regarded as the space of continuous 2$\pi$- (or in this case 1-)periodic functions and in this case Fejér's theorem can be applied. However to use the theorem about equality of measures, one needs to view $\mathbb{T}$ as a metric space. I assume the most natural metric $d$ is defined by ($[x] := x + \mathbb{Z}$) $$ d([x],[y]) = |x-y \text{ mod }1|.$$ for all $[x],[y] \in \mathbb{T}$. Define $f: \mathbb{T} \rightarrow \mathbb{R}$ by $f([x]) = x$ for all $[x] \in \mathbb{T}$, where $x \in [0,1)$ and $x\in [x]$. Then $f$ is continuous with respect to $d$, but the function $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x)=f([x])$ for all $x \in \mathbb{R}$ is certainly not with respect to the euclidean metric. So $C(\mathbb{T})$ as the space of continous function on $\mathbb{T}$ has different functions and the argument above is invalid.
Am I missing something? I'm grateful for any hints, answers and the like.
Instead of defining your metric which turns $\mathbb{T}$ into a metric space which is isometric to $[0,1)$, you should identify $\mathbb{T}$ with the unit circle $S^1 = \{ z \in \mathbb{C} \, | \, |z| = 1 \}$ via the map $\varphi \colon \mathbb{T} \rightarrow S^1$ given by $\varphi([x]) = e^{2\pi i x}$ and pull back the metric from $S^1$ to $\mathbb{T}$ via $\varphi$. Namely, define $$d([x],[y]) := \|\varphi([x]) - \varphi([y])\|$$ where $\| \cdot \|$ is the usual norm on $\mathbb{C}$. Then $\varphi$ becomes an isometry from $\mathbb{T}$ to $S^1$ and a function $g \colon \mathbb{T} \rightarrow \mathbb{R}$ will be continuous with respect to this metric iff there exists a $1$-periodic function $\tilde{g} \colon \mathbb{R} \rightarrow \mathbb{R}$ which is continuous with respect to the usual metrics.