Exercise 2.5

Proof. Let $1<c<3∕2$ be given. Then, for each $m$ let $E_m$ be the open cover of $E$ formed by the union of $2^m$ intervals of length $3^{-m}{c}^m$ covering each segment of $E$ (see Rud76, 2.44). Then, $E\subset {\bigcap <!--\; FUNCTION\; APPLICATION\; -->}_1^{\infty }{E}_m$, but since $\mu (E_m)={(2c∕3)}^m$ for each $m$, we have that $\mu (E)<{(2c∕3)}^m$ for all $m$. Since $1<c<3∕2$, we see $\mu (E)=0$. □