68 1. Measure theory Exercise 1.4.1 (Elementary algebra). Let E[Rd] be the collection of those sets E Rd that are either elementary sets, or co-elementary sets (i.e. the complement of an elementary set). Show that E[Rd] is a Boolean algebra. We will call this algebra the elementary Boolean algebra of Rd. Example 1.4.4 (Jordan algebra). Let J [Rd] be the collection of subsets of Rd that are either Jordan measurable or co-Jordan measurable (i.e. the complement of a Jordan measurable set). Then J [Rd] is a Boolean alge- bra that is finer than the elementary algebra. We refer to this algebra as the Jordan algebra on Rd (but caution that there is a completely different concept of a Jordan algebra in abstract algebra.) Example 1.4.5 (Lebesgue algebra). Let L[Rd] be the collection of Lebesgue measurable subsets of Rd. Then L[Rd] is a Boolean algebra that is finer than the Jordan algebra we refer to this as the Lebesgue algebra on Rd. Example 1.4.6 (Null algebra). Let N (Rd) be the collection of subsets of Rd that are either Lebesgue null sets or Lebesgue co-null sets (the complement of null sets). Then N (Rd) is a Boolean algebra that is coarser than the Lebesgue algebra we refer to it as the null algebra on Rd. Exercise 1.4.2 (Restriction). Let B be a Boolean algebra on a set X, and let Y be a subset of X (not necessarily B-measurable). Show that the restriction B Y := {E Y : E B} of B to Y is a Boolean algebra on Y . If Y is B-measurable, show that B Y = B 2Y = {E Y : E B}. Example 1.4.7 (Atomic algebra). Let X be partitioned into a union X = α∈I of disjoint sets Aα, which we refer to as atoms. Then this partition generates a Boolean algebra A((Aα)α∈I), defined as the collection of all the sets E of the form E = α∈J for some J I, i.e., A((Aα)α∈I) is the collection of all sets that can be represented as the union of one or more atoms. This is easily verified to be a Boolean algebra, and we refer to it as the atomic algebra with atoms (Aα)α∈I. The trivial algebra corresponds to the trivial partition X = X into a single atom at the other extreme, the discrete algebra corresponds to the discrete partition X = x∈X {x} into singleton atoms. More generally, note that finer (resp. coarser) partitions lead to finer (resp. coarser) atomic algebra. In this definition, we permit some of the atoms in the partition to be empty but it is clear that empty atoms have no impact on the final atomic algebra, and so without loss of generality one can delete all empty atoms and assume that all atoms are non-empty if one wishes. Example 1.4.8 (Dyadic algebras). Let n be an integer. The dyadic algebra Dn(Rd) at scale 2−n in Rd is defined to be the atomic algebra generated by
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