TITLE: Connected planar compact set with finite length is path connected
QUESTION [6 upvotes]: Here length means 1-Hausdorff measure. This seems to be known, what is the reference? Or very short proof?

REPLY [7 votes]: See Exercise 3.5 in "The geometry of fractal sets" by K. J. Falconer. 
It says that any such set is an image of rectifiable curve.
For a short proof, check "Rectifiable curve" in my collection.
P.S. The earliest reference I found: Theorem 2 in Continua of finite linear measure. I. by Eilenberg and Harrold (1943).