Introduction to the Mathematical and Statistical Foundations of Econometrics
Properties of Algebras and Sigma-Algebras
1.4.1. General Properties
In this section I will review the most important results regarding algebras, a - algebras, and probability measures.
Our first result is trivial:
Theorem 1.1: If an algebra contains only a finite number of sets, then it is a a-algebra. Consequently, an algebra of subsets of a finite set ^ is a a-algebra.
However, an algebra of subsets of an infinite set ^ is not necessarily a a - algebra. A counterexample is the collection &+ of all subsets of ^ = (0, 1] of the type (a, b], where a < b are rational numbers in [0, 1] together with their finite unions and the empty set 0. Verify that &+ is an algebra. Next, let pn = [10nn]/10n and an = 1/pn, where [x] means truncation to the nearest integer < x. Note that pn n; hence, an I n-1 as n ^ ж. Then, for n = 1, 2, 3,..., (an, 1] є &*, but Uf=1(an, 1] = (n-1, 1] / &+ because n-1 is irrational. Thus, &+ is not a a-algebra.
Theorem 1.2: If& is an algebra, then A, B є & implies A n B є &; hence, by induction, Aj є & for j = 1,...,n < ж implies nn=1 Aj є &. A collection & of subsets of a nonempty set ^ is an algebra if it satisfies condition (1.5) and the condition that, for any pair A, B є &, A n B є &.
Proof: Exercise.
Similarly, we have
Theorem 1.3: If & is a a-algebra, then for any countable sequence of sets Aj є &, Пж=1 Aj є &. A collection & of subsets of a nonempty set ^ is a a-algebra if it satisfies condition (1.5) and the condition that, for any countable sequence of sets Aj є &, Пж=1 Aj є &.
These results will be convenient in cases in which it is easier to prove that (countable) intersections are included in & than to prove that (countable) unions are included:
If & is already an algebra, then condition (1.7) alone would make it a a - algebra. However, the condition in the following theorem is easier to verify than (1.7):
Theorem 1.4: If & is an algebra and Aj, j = 1, 2, 3,... is a countable sequence of sets in &, then there exists a countable sequence of disjoint sets Bj in & such that U°= Aj = U^j=1 Bj. Consequently, an algebra & is also a a-algebra iffor any sequence of disjoint sets Bj in &, U°= Bj e &.
ProofLet Aj e &. Denote B1 = A1, B«+1 = A„+A(U”=1 Aj) = A«+1 П (n”=1 Aj). It follows from the properties of an algebra (see Theorem 1.2) that all the Bj’s are sets in &. Moreover, it is easy to verify that the Bj’s are disjoint and that Aj = Uj= 1 Bj. Thus, if 1 Bj e &, then Aj e &.
Q. E.D.
Theorem 1.5: Let &в, в e ®, be a collection of a - algebras of subsets of a given set ^, where © is a possibly uncountable index set. Then & = ne e_©.& is a a-algebra.
Proof: Exercise.
For example, let &в = {(0, 1], 0, (0, в], (в, 1]}, в e © = (0, 1]. Then Пвe©&e = {(0, 1], 0} is a a-algebra (the trivial a-algebra).
Theorem 1.5 is important because it guarantees that, for any collection ® of subsets of Й, there exists a smallest a-algebra containing ®. By adding complements and countable unions it is possible to extend ® to a a-algebra. This can always be done because ® is contained in the a-algebra of all subsets of Й, but there is often no unique way of doing this except in the case in which ® is finite. Thus, let &в, в e © be the collection of all a-algebras containing ®. Then & = Пвe©&в is the smallest a-algebra containing ®.
Definition 1.4: The smallest a-algebra containing a given collection ® of sets is called the a-algebra generated by ® and is usually denoted by a(Ф).
Note that & = ивe©&в is not always a a-algebra. For example, let ^ = [0, 1] and let, for n > 1, &n = {[0, 1], 0, [0, 1 - n-1], (1 - n-1, 1]}. Then An = [0, 1 - n-1] e &n c Uf=x&n, but the interval [0, 1) = U^An is not contained in any of the a-algebras &n; hence, U))=1 An / U^ &n.
However, it is always possible to extend Uв <.©&в to a a - algebra, often in various ways, by augmenting it with the missing sets. The smallest a-algebra
containing U9 e0 is usually denoted by
v9e0^9 = a (U9Є0^9) .
The notion of smallest a-algebra of subsets of ^ is always relative to a given collection ® of subsets of ^. Without reference to such a given collection ®, the smallest a - algebra of subsets of ^ is {^, 0}, which is called the trivial a-algebra.
Moreover, as in Definition 1.4, we can define the smallest algebra of subsets of ^ containing a given collection ® of subsets of Й, which we will denote by a(@).
For example, let ^ = (0, 1], and let ® be the collection of all intervals of the type (a, b] with 0 < a < b < 1. Then a(@) consists of the sets in ® together with the empty set 0 and all finite unions of disjoint sets in ®. To see this, check first that this collection a(@) is an algebra as follows:
(a) The complement of (a, b] in ® is (0, a] U (b, 1]. If a = 0, then (0, a] = (0, 0] = 0, and if b = 1, then (b, 1] = (1, 1] = 0; hence, (0, a] U (b, 1] is a set in ® or a finite union of disjoint sets in ®.
(b) Let (a, b] in ® and (c, d] in ®, where without loss of generality we may assume that a < c. If b < c, then (a, b] U (c, d] is a union of disjoint sets in ® .If c < b < d, then (a, b] U (c, d] = (a, d] is a set in ® itself, and if b > d, then (a, b] U (c, d] = (a, b] is a set in ® itself. Thus, finite unions of sets in ® are either sets in ® itself or finite unions of disjoint sets in ®.
(c) Let A = Un=j(aj, bj], where 0 < ai < b1 < a2 < b2 < ••• < an < bn < 1. Then A = Un=0(bj, aj+1], where b0 = 0 and an+1 = 1, which is a finite union of disjoint sets in ® itself. Moreover, as in part (b) it is easy to verify that finite unions of sets of the type A can be written as finite unions of disjoint sets in ®.
Thus, the sets in ® together with the empty set 0 and all finite unions of disjoint sets in ® form an algebra of subsets of ^ = (0, 1].
To verify that this is the smallest algebra containing ®, remove one of the sets in this algebra that does not belong to ® itself. Given that all sets in the algebra are of the type A in part (c), let us remove this particular set A. But then Un=1(aj, bj ] is no longer included in the collection; hence, we have to remove each of the intervals (aj, bj ] as well, which, however, is not allowed because they belong to ®.
Note that the algebra a(@) is not a a-algebra because countable infinite unions are not always included in a(®). For example, U^=1(0, 1 - n-1] = (0, 1) is a countable union of sets in a(@), which itself is not included in a(@). However, we can extend a(@) to a(a(@)), the smallest a-algebra containing a(@), which coincides with a(Ф).
