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Good evening!

Let $ \mathbb{T}:=\{ z \in \mathbb{C} ; \vert z \vert =1 \} $ be the unit circle in the complex plane. We denote the trace Borel-$\sigma$-algebra on $\mathbb{T}$ by $\mathcal{B}(\mathbb{T})$.

Here is my question: Does anyone know an elegant method to construct the Haar-measure (in this case the one-dimensional Lebesgue-measure) on $\mathcal{B}(\mathbb{T})$ ?

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    To elaborate on N.S.'s comment, your $\mu = \lambda \circ f^{-1}$ construction would work if you could show that Lebesgue measure $\lambda$ on $[0,1)$ is invariant under translation mod 1. But this is easy if you just note that $A + t \bmod 1 = ((A \cap [0,1-t)) + t) \cup ((A \cap [1-t,1)) - (1-t))$ where the union is disjoint.2012-03-09

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