\(\newcommand{\R}{\mathbb{R}}\)
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There are several other types of algebraic set structures that are weaker that \( \sigma \)-algebras. These are are not particularly important in themselves, but are important for constructing \( \sigma \)-algebras and the measures on these \( \sigma \)-algebras. You may want to skip this section if you are not intersted in questions of existence and uniqueness of positive measures.

Throughout this section, we assume that \( S \) be a set and \( \mathscr{S} \) is a nonempty collection of subsets of \( S \).

\( \mathscr{S} \) is said to be a \( \pi \)-system if \( \mathscr{S} \) is closed under finite intersections: if \( A, \; B \in \mathscr{S} \) then \( A \cap B \in \mathscr{S} \).

Closure under intersection is clearly a very simple property, but \( \pi \) systems turn out to be uesful enough to deserve a name.

\( \mathscr{S} \) is said to be a \( \lambda \)-system if it is closed under complements and countable *disjoint* unions.

- If \( A \in \mathscr{S} \) then \( A^c \in \mathscr{S} \).
- If \( A_i \in \mathscr{S}\) for \( i \) in a countable index set \( I \) and \( A_i \cap A_j = \emptyset \) for \( i \ne j \) then \( \bigcup_{i \in I} A_i \in \mathscr{S} \).

\( \mathscr{S} \) is said to be a semi-algebra if it is closed under intersection and if complements can be written as finite, disjoint unions:

- If \( A, \; B \in \mathscr{S} \) then \( A \cap B \in \mathscr{S} \).
- If \( A \in \mathscr{S} \) then there exists a finite, disjoint collection \( \{B_i: i \in I\} \subseteq \mathscr{S} \) such that \( A^c = \bigcup_{i \in I} B_i \).

For our final structure, recall that a sequence \( (A_1, A_2, \ldots) \) of subsets of \( S \) is increasing if \( A_n \subseteq A_{n+1} \) for all \( n \in \N_+ \). The sequence is decreasing if \( A_{n+1} \subseteq A_n \) for all \( n \in \N_+ \). Of course, these are the standard meanings of *increasing* and *decreasing* relative to the ordinary order \( \le \) on \( \N_+ \) and the subset partial order \( \subseteq \) on \( \mathscr{P}(S) \).

\( \mathscr{S} \) is said to be a monotone class if it is closed under increasing unions and decreasing intersections:

- If \( (A_1, A_2, \ldots) \) is an increasing sequence of sets in \( \mathscr{S} \) then \( \bigcup_{i=1}^\infty A_i \in \mathscr{S} \).
- If \( A_1, A_2, \ldots) \) is a decreasing sequence of sets in \( \mathscr{S} \) then \( \bigcap_{i=1}^\infty A_i \in \mathscr{S} \).

Our most important set structure, the \( \sigma \)-algebra, has all of the properties in the definitions above.

If \( \mathscr{S} \) is a \( \sigma \)-algebra then \( \mathscr{S} \) is a \( \pi \)-system, a \( \lambda \)-system, a semi-algebra, and a monotone class.

If \( \mathscr{S} \) is a \( \lambda \)-system then \( S \in \mathscr{S} \) and \( \emptyset \in \mathscr{S} \).

The proof is just like the one for an algebra. There exists \( A \in \mathscr{S} \) since \( \mathscr{S} \) is non-empty. Hence \( A^c \in \mathscr{S} \) and so \( S = A \cup A^c \in \mathscr{S} \). Finally \( \emptyset = S^c \in \mathscr{S} \).

Any type of algebraic structure on subsets of \( S \) that is defined purely in terms of closure properties will be preserved under intersection. That is, we will have results that are analogous to how \( \sigma \)-algebras are generated from more basic sets, with completely straightforward and analgous proofs. In the following two theorems, the term *system* could mean \( \pi \)-system, \( \lambda \)-system, or monotone class of subsets of \( S \).

If \( \mathscr{S}_i \) is a system for each \( i \) in an index set \( I \) and \( \bigcap_{i \in I} \mathscr{S}_i \) is nonempty, then \( \bigcap_{i \in I} \mathscr{S}_i \) is a system of the same type.

The condition that \( \bigcap_{i \in I} \mathscr{S}_i \) be nonempty is unnecessary for a \( \lambda \)-system, by the result above. Now suppose that \( \mathscr{B} \) is a nonempty collection of subsets of \( S \), thought of as basic sets of some sort. Then the system generated by \( \mathscr{B} \) is the intersection of all systems that contain \( \mathscr{B} \).

The system \( \mathscr{S} \) generated by \( \mathscr{B} \) is the smallest system containing \( \mathscr{B} \), and is characterized by the following properties:

- \( \mathscr{B} \subseteq \mathscr{S} \).
- If \( \mathscr{T} \) is a system and \( \mathscr{B} \subseteq \mathscr{T} \) then \( \mathscr{S} \subseteq \mathscr{T} \).

Note however, that the previous two results do not apply to semi-algebras, because the semi-algebra is not defined purely in terms of closure properties (the condition on \( A^c \) is not a closure property).

If \( \mathscr{S} \) is a monotone class and an algebra, then \( \mathscr{S} \) is a \( \sigma \)-algebra.

All that is needed is to prove closure under countable unions. Thus, suppose that \( A_i \in \mathscr{S} \) for \( i \in \N_+ \). Then \( B_n = \bigcup_{i=1}^n A_i \in \mathscr{S} \) since \( \mathscr{S} \) is an algebra. The sequence \( (B_1, B_2, \ldots) \) is increasing, so \( \bigcup_{n=1}^\infty B_n \in \mathscr{S} \), since \( \mathscr{S} \) is a monotone class. But \( \bigcup_{n=1}^\infty B_n = \bigcup_{i=1}^\infty A_i \).

By definition, a semi-algebra is a \( \pi \)-system. More importantly, a semi-algebra can be used to construct an algebra.

Suppose that \( \mathscr{S} \) is a semi-algebra of subsets of \( S \). Then the collection \( \mathscr{S}^* \) of finite, disjoint unions of sets in \( \mathscr{S} \) is an algebra.

Suppose that \( A, \, B \in \mathscr{S}^* \). Then there exist finite, disjoint collections \( \{A_i: i \in I\} \subseteq \mathscr{S} \) and \( \{B_j: j \in J\} \subseteq \mathscr{S} \) such that \( A = \bigcup_{i \in I} A_i \) and \( B = \bigcup_{j \in J} B_j \). Hence \[ A \cap B = \bigcup_{(i, j) \in I \times J} (A_i \cap B_j) \] But \( \{A_i \cap B_j: (i, j) \in I \times J\} \) is a finite, disjoint collection of sets in \( \mathscr{S} \), so \( A \cap B \in \mathscr{S}^* \). Suppose \( A \in \mathscr{S}^* \), so that there exists a finite, disjoint collection \( \{A_i: i \in I\} \) such that \( A = \bigcup_{i \in I} A_i \). Then \( A^c = \bigcap_{i \in I} A_i^c \). But \( A_i^c \in \mathscr{S}^* \) by definition of semi-algebra, and we just showed that \( \mathscr{S}^* \) is closed under finite intersections, so \( A^c \in \mathscr{S}^* \).

We will say that our nonempty collection \( \mathscr{S} \) is closed under proper set difference if \( A, \; B \in \mathscr{S} \) and \( A \subseteq B \) implies \( B \setminus A \in \mathscr{S} \). The following theorem gives the basic relationship between \( \lambda \) systems and monotone classes.

Suppose that \( \mathscr{S} \) is a nonempty collection of subsets of \( S \).

- If \( \mathscr{S} \) is a \( \lambda \)-system then \( \mathscr{S} \) is a monotone class and is closed under proper set difference.
- If \( \mathscr{S} \) is a monotone class, is closed under proper set difference, and contains \( S \), then \( \mathscr{S} \) is a \( \lambda \)-system.

- Suppose that \( \mathscr{S} \) is a \( \lambda \)-system. Suppose that \( A, \, B \in \mathscr{S} \) and \( A \subseteq B \). Then \( B^c \in \mathscr{S} \), and \( A \) and \( B^c \) are disjoint, so \( A \cup B^c \in \mathscr{S} \). But then \( (A \cup B^c)^c = B \cap A^c = B \setminus A \in \mathscr{S} \). Hence \( \mathscr{S} \) is closed under proper set difference. Next suppose that \( (A_1, A_2, \ldots) \) is an increasing sequence of sets in \( \mathscr{S} \). Let \( B_1 = A_1 \) and \( B_n = A_n \setminus A_{n-1} \) for \( n \in \{2, 3, \ldots\} \). Then \( B_i \in \mathscr{S} \) for each \( i \in \N_+ \). But the sequence \( (B_1, B_2, \ldots) \) is disjoint and has the same union as \( (A_1, A_2, \ldots) \). Hence \( \bigcup_{i=1}^\infty A_i = \bigcup_{i=1}^\infty B_i \in \mathscr{S} \). Finally, suppose that \( (A_1, A_2, \ldots) \) is a decreasing sequence of sets in \( \mathscr{S} \). Then \( A_i^c \in \mathscr{S} \) for each \( i \in \N_+ \) and \( (A_1^c, A_2^c, \ldots) \) is increasing. Hence \( \bigcup_{i=1}^\infty A_i^c \in \mathscr{S} \) and therefore \( \left(\bigcup_{i=1}^\infty A_i^c\right)^c = \bigcap_{i=1}^\infty A_i \in \mathscr{S} \).
- Suppose that \( \mathscr{S} \) is a monotone class, is closed under proper set difference, and \( S \in \mathscr{S} \). If \( A \in \mathscr{S} \) then trivially \( A \subseteq S \) so \( A^c = S \setminus A \in \mathscr{S} \). Next, suppose that \( A, \; B \in \mathscr{S} \) are disjoint. Then \( A^c \in \mathscr{S} \) and \( B \subseteq A^c \), so \( A^c \setminus B = A^c \cap B^c \in \mathscr{S} \). Hence \( A \cup B = (A^c \cap B^c)^c \in \mathscr{S} \). Finally, suppose that \( (A_1, A_2, \ldots) \) is a disjoint sequence of sets in \( \mathscr{S} \). We just showed that \( \mathscr{S} \) is closed under finite, disjoint unions, so \( B_n = \bigcup_{i=1}^n A_i \in \mathscr{S} \). But the sequence \( (B_1, B_2, \ldots) \) is increasing, and hence \( \bigcup_{n=1}^\infty B_n = \bigcup_{i=1}^\infty A_i \in \mathscr{S} \).

The following theorem is known as the montone class theorem, and is due to the mathematician Paul Halmos.

Suppose that \( \mathscr{A} \) is an algebra, \( \mathscr{M} \) is a montone class, and \( \mathscr{A} \subseteq \mathscr{M} \). Then \( \sigma(\mathscr{A}) \subseteq \mathscr{M} \).

First let \( m(\mathscr{A}) \) denote the monotone class generated by \( \mathscr{A} \), as defined above. The outline of the proof is to show that \( m(\mathscr{A}) \) is an algebra, so that by (9), \( m(\mathscr{A}) \) is a \( \sigma \)-algebra. It then follows that \( \sigma(\mathscr{A}) \subseteq m(\mathscr{A}) \subseteq \mathscr{M} \). To show that \( m(\mathscr{A}) \) is an algebra, we first show that it is closed under complements and then under simple union.

Since \( m(\mathscr{A}) \) is a montone class, the collection \( m^*(\mathscr{A}) = \{A \subseteq S: A^c \in m(\mathscr{A})\} \) is also a montone class. Moreover, \( \mathscr{A} \subseteq m^*(\mathscr{A}) \) so it follows that \( m(\mathscr{A}) \subseteq m^*(\mathscr{A}) \). Hence if \( A \in m(\mathscr{A}) \) then \( A \in m^*(\mathscr{A}) \) so \( A^c \in m(\mathscr{A}) \). Thus \( m(\mathscr{A}) \) is closed under complements.

Let \( \mathscr{M}_1 = \{A \subseteq S: A \cup B \in m(\mathscr{A}) \text{ for all } B \in \mathscr{A}\} \). Then \( \mathscr{M}_1 \) is a monotone class and \( \mathscr{A} \subseteq \mathscr{M}_1 \) so \( m(\mathscr{A}) \subseteq \mathscr{M}_1 \). Next let \( \mathscr{M}_2 = \{A \subseteq S: A \cup B \in m(\mathscr{A}) \text{ for all } B \in m(\mathscr{A})\} \). Then \( \mathscr{M}_2 \) is also a montone class. Let \( A \in \mathscr{A} \). If \( B \in m(\mathscr{A})\) then \( B \in \mathscr{M}_1 \) and hence \( A \cup B \in m(\mathscr{A}) \). Hence \( A \in \mathscr{M}_2 \). Thus we have \( \mathscr{A} \subseteq \mathscr{M}_2 \), so \( m(\mathscr{A}) \subseteq \mathscr{M}_2 \). Finally, let \( A, \, B \in m(\mathscr{A}) \). Then \( A \in \mathscr{M}_2 \) so \( A \cup B \in m(\mathscr{A}) \) and therefore \( m(\mathscr{A}) \) is closed under simple union.

As noted above, a \( \sigma \)-algebra is both a \( \pi \)-system and a \( \lambda \)-system. The converse is also true, and is one of the main reasons for studying these structures.

If \( \mathscr{S} \) is a \( \pi \)-system and a \( \lambda \)-system then \( \mathscr{S} \) is a \( \sigma \)-algebra.

\( S \in \mathscr{S} \), and if \( A \in \mathscr{S} \) then \( A^c \in \mathscr{S} \) by definition of a \( \lambda \)-system. Thus, all that is left is to show closure under countable unions. Thus, suppose that \( (A_1, A_2, \ldots) \) is a sequence of sets in \( \mathscr{S} \). Then \( A_i^c \in \mathscr{S} \) for each \( i \in \N_+ \). Since \( \mathscr{S} \) is also a \( \pi \)-system, it follows for for each \( n \in \N_+ \), \( B_n = A_n \cap A_1^c \cap \cdots \cap A_{n-1}^c \in \mathscr{S} \) (by convenction \( B_1 = A_1 \)). But the sequence \( (B_1, B_2, \ldots) \) is disjoint and has the same union as \( (A_1, A_2, \ldots) \). Hence \( \bigcup_{i=1}^\infty A_i = \bigcup_{i=1}^\infty B_i \in \mathscr{S} \).

The importance of \( \pi \)-systems and \( \lambda \)-systems stems in part from Dynkin's \( \pi \)-\( \lambda \) theorem given next. It's named for the mathematician Eugene Dynkin.

Suppose that \( \mathscr{A} \) is a \( \pi \)-system of subsets of \( S \), \( \mathscr{B} \) is a \( \lambda \)-system of subsets of \( S \), and \( \mathscr{A} \subseteq \mathscr{B} \). Then \( \sigma(\mathscr{A}) \subseteq \mathscr{B} \).

Let \( \mathscr{L} \) denote the \( \lambda \)-system generated by \( \mathscr{A} \). Then of course \( \mathscr{A} \subseteq \mathscr{L} \subseteq \mathscr{B} \). For \( A \in \mathscr{L} \), let \[ \mathscr{L}_A = \{B \subseteq S: B \cap A \in \mathscr{L}\}\] We will show that \( \mathscr{L}_A \) is a \( \lambda \)-system. Note that \( S \cap A = A \in \mathscr{L}\) and therefore \( S \in \mathscr{L}_A \). Next, suppose that \( B_1, \, B_2 \in \mathscr{L}_A \) and that \( B_1 \subseteq B_2 \). Then \( B_1 \cap A \in \mathscr{L} \) and \( B_1 \cap A \in \mathscr{L} \) and \( B_1 \cap A \subseteq B_2 \cap A \). Hence \( (B_2 \setminus B_1) \cap A = (B_2 \cap A) \setminus (B_1 \cap A) \in \mathscr{L} \). Hence \( B_2 \setminus B_1 \in \mathscr{L}_A \). Finally, suppose that \( \{B_i: i \in I\} \) is a countable, disjoint collection of sets in \( \mathscr{L}_A \). Then \( B_i \cap A \in \mathscr{L} \) for each \( i \in I \), and \( \{B_i \cap A: i \in I\} \) is also a disjoint collection. Therefore, \( \bigcup_{i \in I} (B_i \cap A) = \left(\bigcup_{i \in I} B_i \right) \cap A \in \mathscr{L} \). Hence \( \bigcup_{i \in I} B_i \in \mathscr{L}_A \).

Next fix \( A \in \mathscr{A} \). If \( B \in \mathscr{A} \) then \( A \cap B \in \mathscr{A} \), so \( A \cap B \in \mathscr{L} \) and hence \( B \in \mathscr{L}_A \). But \( \mathscr{L} \) is the smallest \( \lambda \)-system containing \( \mathscr{A} \) so we have shown that \( \mathscr{L} \subseteq \mathscr{L}_A \) for every \( A \in \mathscr{A} \). Now fix \( B \in \mathscr{L}\). If \( A \in \mathscr{A} \) then \( B \in \mathscr{L}_A \) so \( A \cap B \in \mathscr{L} \) and therefore \( A \in \mathscr{L}_B \). Again, \( \mathscr{L} \) is the smallest \( \lambda \)-system containing \( \mathscr{A} \) so we have now shown that \( \mathscr{L} \subseteq \mathscr{L}_B \) for every \( B \in \mathscr{L} \). Finally, let \( B, \, C \in \mathscr{L} \). Then \( C \in \mathscr{L}_B \) and hence \( B \cap C \in \mathscr{L} \). It now follows that \( \mathscr{L} \) is a \( \pi \)-system, as well as a \( \lambda \)-system, and therefore by the previous theorem, \( \mathscr{L} \) is a sigma-algebra. But \( \mathscr{A} \subseteq \mathscr{L} \) and hence \( \sigma(\mathscr{A}) \subseteq \mathscr{L} \).

Suppose that \( S \) is a set and \( \mathscr{A} \) is a finte partition of \( S \). Then \( \mathscr{S} = \{\emptyset\} \cup \mathscr{A} \) is a semi-algebra of subsets of \( S \).

If \( A, \, B \in \mathscr{A} \) then \( A \cap B = \emptyset \in \mathscr{S} \). If \( A \in \mathscr{S} \) then \( A^c = \bigcup\{B \in \mathscr{A}: B \neq A \}\)

The following example is particulalry important because it will be used to construct positive measures on \( \R \). Let \[ \mathscr{B} = \{(a, b]: a, \, b \in \R, \; a \lt b\} \cup \{(-\infty, b]: b \in \R\} \cup \{(a, \infty): a \in \R \} \]

\( \mathscr{B} \) is a semi-algebra of subsets of \( \R \).

Note that the intersection of two intervals of the type in \( \mathscr{B} \) is another interval of this type. The complement of an interval of this type is either another interval of this type or the union of two disjoint intervals of this type.

It follows from result above that the collection \( \mathscr{A} \) of finite disjoint unions of intervals in \( \mathscr{B} \) is an algebra. Recall also that \( \sigma(\mathscr{B}) = \sigma(\mathscr{A}) \) is the Borel \( \sigma \)-algebra of \( \R \), named for Émile Borel. We can generalize all of this to \( \R^n \) for \( n \in \N_+ \)

The collection \( \mathscr{B}_n = \left\{A_1 \times A_2 \times \cdots \times A_n: A_i \in \mathscr{B} \text{ for each } i \in \{1, 2, \ldots n\} \right\} \) is a semi-algebra of subsets of \( \R^n \).

Recall also that \( \sigma(\mathscr{B}_n) \) is the \( \sigma \)-algebra of Borel sets of \( \R^n \).