Show that any disjoint intervals is countable.
Show that every open \(U \subseteq \RR\) is a countable union of disjoint open intervals.
Show that every open \(U \subseteq \RR^n\) is a countable union of almost disjoint closed cubes.
Show that that Cantor middle-thirds set is compact, totally disconnected, and perfect, with outer measure zero.
Prove the Borel-Cantelli lemma.
Solution
1. A family of pairwise disjoint intervals in \(\RR\), each with nonempty interior, is countable.
Each interval contains a rational number; choosing one for each interval gives an injection into \(\QQ\), because the intervals are disjoint. See E-MFCCS, which also shows that the family of singletons \(\theset{x}\), \(x \in \RR\), is an uncountable disjoint family of degenerate intervals.
2. Every open \(U \subseteq \RR\) is a countable union of disjoint open intervals.
The connected components of \(U\) partition \(U\). Each is open, because \(\RR\) is locally connected, and connected, hence an open interval. By step 1 there are countably many. See E-EJN4Q.
3. Every open \(U \subseteq \RR^n\) is a countable union of almost disjoint closed cubes.
Let \(\mathcal F\) be the set of closed dyadic cubes \(Q \subseteq U\) of side \(2^{-k}\), \(k \geq 0\), such that \(k = 0\) or the dyadic cube of side \(2^{-k+1}\) containing \(Q\) is not contained in \(U\). Every \(x \in U\) lies in a dyadic cube inside \(U\), and the one of largest side among the dyadic cubes containing \(x\) of side at most \(1\) and contained in \(U\) lies in \(\mathcal F\). Two dyadic cubes are nested or have disjoint interiors, and the defining condition rules out strict nesting within \(\mathcal F\). There are countably many dyadic cubes. See E-M3CGY.
4. The Cantor set \(C = \bigcap_n C_n\), with \(C_n\) the union of the \(2^n\) closed intervals of length \(3^{-n}\) remaining at stage \(n\), is compact, of outer measure zero, totally disconnected, and perfect.
\(C\) is a closed subset of \([0,1]\), hence compact, and \(m^*(C) \le m(C_n) = (2/3)^n\) for every \(n\). A connected subset of \(\RR\) with two points contains an interval of positive length, which would have positive measure, so every connected subset of \(C\) has at most one point. For \(x \in C\) and each \(n\), the stage-\(n\) interval containing \(x\) has two endpoints in \(C\) at distance \(3^{-n}\) apart, so one of them is a point of \(C \setminus \theset{x}\) within \(3^{-n}\) of \(x\). See E-L3F4O.
5. If \((E_n)\) are measurable with \(\sum_n \mu(E_n) < \infty\), then \(\mu(\limsup_n E_n) = 0\).
\(\limsup_n E_n = \bigcap_N\bigcup_{n \ge N}E_n\) is measurable and contained in \(\bigcup_{n \ge N}E_n\) for every \(N\), so countable subadditivity gives \(\mu(\limsup_n E_n) \le \sum_{n \ge N}\mu(E_n)\). The right side is the tail of a convergent series and tends to \(0\).