Theorem 9.1
If \(O\) is a bounded open set and \(p\) is a point of \(O\) then there is a unique open interval containing \(p\) which is a subset of \(O\) whose endpoints do not lie in \(O\text{.}\)
Chapter 9 Measure Theory
Now that we know that the intervals described in Theorem 9.1 exist, we can formally define them and give them a name.
Definition 9.2
If \(O\) is a bounded open set and \(p \in O\) then the open interval containing \(p\) which is a subset of \(O\) and whose endpoints are not in \(O\) is called the component of \(O\) containing \(p\).
Theorem 9.3
If \(O\) is a bounded open set, then the set of all components of \(O\) is a countable collection of mutually disjoint open intervals whose union is \(O\text{.}\)
Definition 9.4
If \(S\) is any interval (open, closed, or half-open), then we define \(L(S)\) to be the length of \(S\). For example, if \(S = [a,b],\) then \(L(S)= b-a\text{.}\) If \(G\) is a finite collection of mutually disjoint open intervals then \(L(G)\) denotes the sum of the lengths of the elements of \(G\text{.}\) If \(G = \{g_1, g_2, g_3, \dots \}\) is a countable collection of mutually disjoint open intervals lying in an open interval, then \(L(G) = \sum_{i=1}^\infty L(g_i)\text{.}\)
Theorem 9.5
If \(G\) is a finite collection of mutually disjoint open intervals lying in the open interval \((a,b)\text{,}\) then \(L(G) \leq b-a\text{.}\)
Theorem 9.6
If \(G\) is a countable collection of mutually disjoint open intervals lying in the open interval \((a,b)\) then \(L(G) \leq b-a\text{.}\)
Definition 9.7
If \(G\) is a collection of point sets then \(G^*\) denotes the set which is the union of the members of \(G\text{;}\) that is, \(G^* = \bigcup_{g \in G} g\text{.}\)
Theorem 9.8
If \(G\) and \(H\) are countable collections of mutually disjoint open intervals and \(G^* \subseteq H^*\text{,}\) then \(L(G) \leq L(H)\text{.}\)
Definition 9.9
A point set \(M\) is said to be closed if and only if no point not in \(M\) is a limit point of \(M\text{.}\)
Theorem 9.10
If \(O\) is an open set which is a subset of the closed interval \([a,b]\text{,}\) then \([a,b]-O\) is a closed point set, and if \(M\) is a closed point set which lies in an open interval \((a,b)\text{,}\) then \((a,b)-M\) is an open set.
Definition 9.11
The statement that the set \(G\) of open intervals properly covers the set \(M\) means that every point of \(M\) lies in a member of \(G\) and every member of \(G\) contains a point of \(M\text{.}\)
Definition 9.12
If \(M\) is a bounded point set then by the outer measure of \(M\), denoted \(m_o(M),\) is meant the greatest lower bound of the set of all \(L(G)\) where \(G\) is any collection of mutually disjoint open intervals which properly cover \(M\text{.}\)
Theorem 9.13
Show that if \(M\) is a countable point set then the outer measure of \(M\) is zero.
Theorem 9.14
If \(O\) is a bounded open set and \(G\) is the set of all components of \(O\text{,}\) then \(m_o(O)=L(G)\text{.}\)
Theorem 9.15
If \([a,b]\) is a closed interval, then \(m_o([a,b])=b-a\text{.}\)
Theorem 9.16
If \(M\) is a bounded point set and \(I\) and \(J\) are two closed intervals containing \(M\text{,}\) and \(I \subset J\text{,}\) then \(m_o(I)-m_o(I-M) = m_o(J)-m_o(J-M)\text{.}\)
Theorem 9.17
If \(M\) is a bounded point set and \(I\) and \(J\) are two closed intervals containing \(M\text{,}\) then \(m_o(I)-m_o(I-M) = m_o(J)-m_o(J-M)\text{.}\)
Definition 9.18
If \(M\) is a bounded point set, then the inner measure of \(M\), denoted \(m_i(M),\) means \(m_o(I)-m_o(I-M)\) for some closed interval, \(I\text{,}\) containing \(M\text{.}\)
Definition 9.19
The statement that the point set \(M\) is measurable means that \(m_o(M) = m_i(M)\text{.}\) When \(M\) is measurable, \(m_o(M)\) is called the measure of \(M\) and denoted by \(m(M)\text{.}\)
Theorem 9.20
If \(M\) is a bounded, measurable set, and \(I\) is a closed interval containing \(M\text{,}\) then \(I-M\) is measurable.
Theorem 9.21
If \(H\) and \(K\) are disjoint, bounded sets, then \(m_o(H \cup K) \leq m_o(H) + m_o(K)\text{.}\)
Theorem 9.22
If \(H\) and \(K\) are disjoint, bounded measurable sets then \(H \cup K\) is measurable.
Theorem 9.23
If \(H\) and \(K\) are disjoint, bounded measurable sets then \(m(H \cup K) = m(H) + m(K)\text{.}\)
Theorem 9.24
If \(M\) is a closed interval or an open interval, then \(M\) is measurable and \(L(M)~=~m(M)\text{.}\)
Theorem 9.25
If \(G\) is a collection of open intervals covering the closed interval \([a,b]\) then there is a finite subcollection of \(G\) which also covers \([a,b]\text{.}\)