Exercise 1.1.12 (Jordan null sets)

Let $E$ be a Jordan null set $m_J(E)=0$. And let $F\subset E$. We want to show that there exist covers $A,B:A\subseteq F\subseteq B$ such that $m(A),m(A),m(B)\le 𝜖$.

For the outer cover $B$ this is obvious, since we can simply find a cover $B$ of $E$ with $m(B)\le 𝜖$ by definition, and by $F\subseteq E\subseteq B$ we have found an outer cover of $F$ with measure $𝜖$-close to 0.

For the inner cover $A$ it is actually also not that complicated, since any inner cover $A\subseteq F$ is also an inner cover of $A\subseteq E$; thus, $0\le m(A)\le m_{\ast ,(J)}(E)=0$, i.e, any inner cover of $F$ must have measure zero.

Thus, we can conclude that $m_{\ast ,(J)}(F)=m^{\ast ,(J)}(F)=0$, as desired.