Is the Lebesgue measure of the unit circle $0$? I have a two-dimensional Lebesgue integral over $\partial D(0,1)$. I think it should be zero because the two-dimensional measure of the unit circle is $0$. 
But I'm not sure how to justify this. My idea is to cover the unit circle by countably many boxes of area $\epsilon/2^n$ but I'm not sure how to go about this. Can you please help?
 A: For a very nice region, the Lebesgue measure in the two-dimensional space measures the area of the region.
Consider the region consisting on the points $S_\epsilon$ between the circle of radius $1-\epsilon$ and $1+\epsilon$. The area of this region is
$$\begin{align*}
S_\epsilon&=\pi\cdot (1+\epsilon)^2 - \pi\cdot (1-\epsilon)^2\\
&=\pi\cdot [(1+\epsilon)+(1-\epsilon]\cdot [(1+\epsilon)-(1-\epsilon)]\\
&=4\pi\cdot \epsilon 
\end{align*}$$
Since the circle of radius $1$ (which is usually denoted by $S^1$) is contained in each $S_\epsilon$, we know that 
$$0\leq \lambda(S^1)\leq \lim_{\epsilon\to 0}\lambda (S_\epsilon)=\lim_{\epsilon\to 0}4\pi\cdot \epsilon=0.$$
Thus, $\lambda(S^1)=0$.
A: Cover the circle by annuli of arbitrarily small width.
A: You can approximate the circle with a sequence of inscribed regular n-gons and another sequence of regular n-gons such that the circle is inscribed into them. The limiting measures for the two sequences will be the same, thus the measure of the circle itself will be 0.
