1.4. Abstract measure spaces 69 the half-open dyadic cubes i1 2n , i1 + 1 2n × . . . × id 2n , id + 1 2n of length 2−n (see Exercise 1.1.14). These are Boolean algebras which are increasing in n: Dn+1 Dn. Draw a diagram to indicate how these algebras sit in relation to the elementary, Jordan, and Lebesgue, null, discrete, and trivial algebras. Remark 1.4.9. The dyadic algebras are analogous to the finite resolution one has on modern computer monitors, which subdivide space into square pixels. A low resolution monitor (in which each pixel has a large size) can only resolve a very small set of “blocky” images, as opposed to the larger class of images that can be resolved by a finer resolution monitor. Exercise 1.4.3. Show that the non-empty atoms of an atomic algebra are determined up to relabeling. More precisely, show that if X = α∈I = α ∈I are two partitions of X into non-empty atoms Aα, , then A((Aα)α∈I ) = A((Aα ∈I ) if and only if there exists a bijection φ: I I such that Aφ(α) = for all α I. While many Boolean algebras are atomic, many are not, as the following two exercises indicate. Exercise 1.4.4. Show that every finite Boolean algebra is an atomic al- gebra. (A Boolean algebra B is finite if its cardinality is finite, i.e., there are only finitely many measurable sets.) Conclude that every finite Boolean algebra has a cardinality of the form 2n for some natural number n. From this exercise and Exercise 1.4.3 we see that there is a one-to-one correspon- dence between finite Boolean algebras on X and finite partitions of X into non-empty sets (up to relabeling). Exercise 1.4.5. Show that the elementary, Jordan, Lebesgue, and null algebras are not atomic algebras. (Hint: Argue by contradiction. If these algebras were atomic, what must the atoms be?) Now we describe some further ways to generate Boolean algebras. Exercise 1.4.6 (Intersection of algebras). Let (Bα)α∈I be a family of Boolean algebras on a set X, indexed by a (possibly infinite or uncountable) label set I. Show that the intersection α∈I := α∈I of these algebras is still a Boolean algebra, and is the finest Boolean algebra that is coarser than all Bα. (If I is empty, we adopt the convention that α∈I is the discrete algebra.) Definition 1.4.10 (Generation of algebras). Let F be any family of sets in X. We define Fbool to be the intersection of all the Boolean algebras that
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