MathDB

Let

K\subseteq \mathbb{R}

be compact. Prove that the following two statements are equivalent to each other. (a) For each point

x

K

we can assign an uncountable set

F_x\subseteq \mathbb{R}

such that

\mathrm{dist}(F_x, F_y)\ge |x-y|

holds for all

x,y\in K

; (b)

K

is of measure zero.

6