I'm aware that Lebesgue-measurability captures $n-$dimensional volume in $\mathbb{R}^n$, so curves for example have Lebesgue measure 0.
My basic question is: what captures the 1-dimensional volume of lines and curves in $\mathbb{R}^2$? Is this simply done by "flattening" the curve and analyzing its measurability in $\mathbb{R}$? I'm very new to this stuff so forgive me if I'm very off the mark. I ultimately want to describe what subsets of the unit circle have a defined length.
Best Answer
This depends for starters on what you mean by "curve". If by curve you mean a continuously differentiable map $\gamma:[0,1]\to\mathbb R^n$ then yes, the Lebesgue measure is zero when $n>1$. But in such case you can calculate the length by $$ L(\gamma)=\int_0^1\|\gamma'(t)\|\,dt. $$
If you allow "curves" that are continuous maps $\gamma:[0,1]\to\mathbb R^n$, then it is not true that they necessarily have zero Lebesgue measure. For $n=2$ there is the famous Hilbert Curve, where $m(\gamma([0,1]))=1$.
With the unit circle, the approach is that as long as you can remove a point, it is homeomorphic (diffeomorphic too) with a segment. So you can use the usual Lebesgue measure from the line.