Let \(E \subseteq \RR^n\) be bounded. Prove the following are equivalent:
For any \(\epsilon>0\) there exists and open set \(G\) and a closed set \(F\) such that \[ F \subseteq E \subseteq G && m(G\sm F) < \epsilon .\]
There exists a \(G_ \delta\) set \(V\) and an \(F_ \sigma\) set \(H\) such that \[ m(V\sm H) = 0 .\]
Erratum: condition (2) needs \(H \subseteq E \subseteq V\). As printed, it holds for every \(E\) with \(V = H = \emptyset\), while (1) fails for a bounded non-measurable \(E\).
Solution
Condition (2) is read with \(H \subseteq E \subseteq V\); see the remark below. Fix an open ball \(B \supseteq E\).
1. (1) implies (2).
For each \(k\), (1) with \(\eps = 1/k\) gives open \(G_k\) and closed \(F_k\) with \(F_k \subseteq E \subseteq G_k\) and \(m(G_k \setminus F_k) < 1/k\). Put \(V = \bigcap_k G_k\), a \(G_\delta\) set, and \(H = \bigcup_k F_k\), an \(F_\sigma\) set. Then \(H \subseteq E \subseteq V\), and \(V \setminus H \subseteq G_k \setminus F_k\) for each \(k\), so \(m(V \setminus H) < 1/k\) for every \(k\).
2. (2) implies (1).
2.1. There are open \(G_1 \supseteq G_2 \supseteq \cdots\) with \(\bigcap_m G_m = V \cap B\) and \(m(G_1) < \infty\), and closed \(F_1 \subseteq F_2 \subseteq \cdots\) with \(\bigcup_m F_m = H\).
Write \(V = \bigcap_j U_j\) with \(U_j\) open and \(H = \bigcup_j K_j\) with \(K_j\) closed, and put \(G_m = B \cap U_1 \cap \cdots \cap U_m\) and \(F_m = K_1 \cup \cdots \cup K_m\). Then \(G_1 \subseteq B\) has finite measure. Since \(E \subseteq B\), \(E \subseteq V \cap B\).
2.2. \(m(G_m \setminus (V \cap B)) \to 0\) and \(m(H \setminus F_m) \to 0\).
The first is continuity from above for \(G_m \downarrow V \cap B\), valid because \(m(G_1) < \infty\). The second is continuity from below for \(F_m \uparrow H\), with \(m(H) \le m(B) < \infty\) because \(H \subseteq E\).
2.3. Q.E.D.
Given \(\eps > 0\), step 2 gives \(m\) with \(m(G_m \setminus (V\cap B)) < \eps/2\) and \(m(H \setminus F_m) < \eps/2\). Put \(G = G_m\) and \(F = F_m\). Then \(F \subseteq H \subseteq E \subseteq V \cap B \subseteq G\), and \[ m(G \setminus F) \le m(G \setminus (V\cap B)) + m(V \setminus H) + m(H \setminus F) < \eps. \]
3. Q.E.D.
Steps 1 and 2.