Lebesgue measure - natural measure on $S^1$ I know the definition of the Lebesgue measure for subsets of $\mathbb{R}$. I have heard that the Lebesgue measure is the "natural" measure to put on the circle, $S^1$. Why is this so? 
How is the Lebesgue measure defined for $S^1$ in $\mathbb{R}^2$ or $\mathbb{C}$? Is the idea to take the infimum of the area of open boxes covering $S^1$? If so, how exactly does this work, and what is the Lebesgue measure of $S^1$?

The Lebesgue measure on $S^1$ can be viewed as the Hausdorff measure of it as a subset of $R^2$ with the 2D Lebesgue measure $m_2$. Alternately it can be viewed as the pushfoward measure $\mu(A)=m_1(f^{-1}(A))$ where $f(t)=(\cos(t),\sin(t)$ with domain $[0,2\pi)$. Either way, the measure of an arc is the length of the arc, which makes sense.
