70 1. Measure theory contain F, which is again a Boolean algebra by Exercise 1.4.6. Equivalently, Fbool is the coarsest Boolean algebra that contains F. We say that Fbool is the Boolean algebra generated by F. Example 1.4.11. F is a Boolean algebra if and only if Fbool = F thus each Boolean algebra is generated by itself. Exercise 1.4.7. Show that the elementary algebra E(Rd) is generated by the collection of boxes in Rd. Exercise 1.4.8. Let n be a natural number. Show that if F is a finite collection of n sets, then Fbool is a finite Boolean algebra of cardinality at most 22n (in particular, finite sets generate finite algebras). Give an example to show that this bound is best possible. (Hint: For the latter, it may be convenient to use a discrete ambient space such as the discrete cube X = {0, 1}n.) The Boolean algebra Fbool can be described explicitly in terms of F as follows: Exercise 1.4.9 (Recursive description of a generated Boolean algebra). Let F be a collection of sets in a set X. Define the sets F0, F1, F2,... recursively as follows: (i) F0 := F. (ii) For each n ≥ 1, we define Fn to be the collection of all sets that either the union of a finite number of sets in Fn−1 (including the empty union ∅), or the complement of such a union. Show that Fbool = ∞ n=0 Fn. 1.4.2. σ-algebras and measurable spaces. In order to obtain a measure and integration theory that can cope well with limits, the finite union axiom of a Boolean algebra is insuﬃcient, and must be improved to a countable union axiom: Definition 1.4.12 (Sigma algebras). Let X be a set. A σ-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) (Countable unions) If E1,E2,... ∈ B, then ∞ n=1 En ∈ B. We refer to the pair (X, B) of a set X together with a σ-algebra on that set as a measurable space.

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