In Conways text a course in operator theory he mentions the normalised Lebesgue measure on the unit cirle $\partial\mathbb{D}$of the complex plane. But there is no definition in the text.Here is what I thought.
The map $t\mapsto e^{it} = cos t + i sin t$ can be used to identify the interval (−π, π] with $\partial\mathbb{D}$. We use this map to define a σ-algebra on $\partial\mathbb{D}$ by transferring the Borel subsets of (−π, π] to subsets of $\partial\mathbb{D}$. More precisely, A subset $E$ of $\partial\mathbb{D}$ is defined to be measurable iff $\{t\in (−π, π] : e^{it} ∈ E\}$ is a Borel subset of $\mathbb{R}$.
Then we transfer Lebesgue measure on the Borel subsets of (−π, π] to a measure $\sigma$ on the measurable subsets of $\partial\mathbb{D}$, except that we normalise by dividing by $2\pi$ so that the measure of $\partial\mathbb{D}$ is $1$ rather than $2\pi$. More precisely, if $E\subseteq\partial\mathbb{D}$ is measurable, then $\sigma(E) = \frac{|\{t ∈ (−\pi,\pi]:e^{it}\in E\}|}{2\pi}$.
Is my definition of the normalised Lebesgue measure on the unit circle correct?