A weird classification of compact subset of R of zero measure

From: Lukas Horosiewicz (horosiewicz_at_gmail.com)
Date: 10/30/04 

Date: 30 Oct 2004 10:50:29 -0700

If K \subset \mathbb{R} is compact how does one see that the following
condictions are equivalent:

(1) For all x \in K we can assign an uncountable set F_x in \mathbb{R}
such that |x - y| \leq d(F_x,F_y) for all x,y \in K.
(2) K is of 0 measure.

I have no idea of even where to start.