1.4. Abstract measure spaces 67 1.4.1. Boolean algebras. We begin by recalling the concept of a Boolean algebra. Definition 1.4.1 (Boolean algebras). Let X be a set. A (concrete) Boolean algebra on X is a collection B of X which obeys the following properties: (i) (Empty set) B. (ii) (Complement) If E B, then the complement Ec := X\E also lies in B. (iii) (Finite unions) If E, F B, then E F B. We sometimes say that E is B-measurable, or measurable with respect to B, if E B. Given two Boolean algebras B, B on X, we say that B is finer than, a sub-algebra of, or a refinement of B, or that B is coarser than or a coarsening of B , if B B . We have chosen a “minimalist” definition of a Boolean algebra, in which one is only assumed to be closed under two of the basic Boolean operations, namely complement and finite union. However, by using the laws of Boolean algebra (such as de Morgan’s laws), it is easy to see that a Boolean algebra is also closed under other Boolean algebra operations such as intersection E F , set differerence E\F , and symmetric difference EΔF . So one could have placed these additional closure properties inside the definition of a Boolean algebra without any loss of generality. However, when we are verifying that a given collection B of sets is indeed a Boolean algebra, it is convenient to have as minimal a set of axioms as possible. Remark 1.4.2. One can also consider abstract Boolean algebras B, which do not necessarily live in an ambient domain X, but for which one has a collection of abstract Boolean operations such as meet and join instead of the concrete operations of intersection and union ∪. We will not take this abstract perspective here, but see §2.3 of An epsilon of room, Vol. I for some further discussion of the relationship between concrete and abstract Boolean algebras, which is codified by Stone’s theorem. Example 1.4.3 (Trivial and discrete algebra). Given any set X, the coars- est Boolean algebra is the trivial algebra {∅,X}, in which the only measur- able sets are the empty set and the whole set. The finest Boolean algebra is the discrete algebra 2X := {E : E X}, in which every set is measurable. All other Boolean algebras are intermediate between these two extremes: finer than the trivial algebra, but coarser than the discrete one.
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