Rectifiable set

Remarks

This is the countable rectifiability of Federer and Mattila. Some authors also require $\mathcal{H}^m(E) < \infty$, or use $C^1$ maps instead of Lipschitz maps; the second change gives the same class of sets. Example: the unit circle $S \subseteq \R^2$ is $1$-rectifiable, since the Lipschitz map $f(t) = (\cos t, \sin t)$ satisfies $f(\R) = S$. Non-example: the unit square $Q = [0,1]^2$ is not $1$-rectifiable. Each set $f_i(\R)$ has $\sigma$-finite $\mathcal{H}^1$ measure, so it has Lebesgue measure zero in $\R^2$. Hence $Q \setminus \bigcup_i f_i(\R)$ has positive area, and so its $\mathcal{H}^1$ measure is infinite, not zero. The image of a rectifiable curve is $1$-rectifiable.