Finite curvature of arc length measure implies rectifiability: a new proof Xavier Tolsa 30C8528A75curvature of measuresrectifiabilityanalytic capacity If $E\subset\mathbb{C}$ is a set with finite length and finite curvature, then $E$ is rectifiable. This fact, proved by David and L\'eger in 1999, is one of the basic ingredients for the proof of Vitushkin's conjecture. In this paper we give another different proof of this result. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2746 10.1512/iumj.2005.54.2746 en Indiana Univ. Math. J. 54 (2005) 1075 - 1106 state-of-the-art mathematics http://iumj.org/access/