1.4. Abstract measure spaces 71 Remark 1.4.13. The prefix σ usually denotes “countable union”. Other instances of this prefix include a σ-compact topological space (a countable union of compact sets), a σ-finite measure space (a countable union of sets of finite measure), or set (a countable union of closed sets) for other instances of this prefix. From de Morgan’s law (which is just as valid for infinite unions and intersections as it is for finite ones), we see that σ-algebras are closed under countable intersections as well as countable unions. By padding a finite union into a countable union by using the empty set, we see that every σ-algebra is automatically a Boolean algebra. Thus, we automatically inherit the notion of being measurable with respect to a σ-algebra, or of one σ-algebra being coarser or finer than another. Exercise 1.4.10. Show that all atomic algebras are σ-algebras. In partic- ular, the discrete algebra and trivial algebra are σ-algebras, as are the finite algebras and the dyadic algebras on Euclidean spaces. Exercise 1.4.11. Show that the Lebesgue and null algebras are σ-algebras, but the elementary and Jordan algebras are not. Exercise 1.4.12. Show that any restriction B Y of a σ-algebra B to a subspace Y of X (as defined in Exercise 1.4.2) is again a σ-algebra on the subspace Y . There is an exact analogue of Exercise 1.4.6: Exercise 1.4.13 (Intersection of σ-algebras). Show that the intersection α∈I := α∈I of an arbitrary (and possibly infinite or uncountable) number of σ-algebras is again a σ-algebra, and is the finest σ-algebra that is coarser than all of the Bα. Similarly, we have a notion of generation: Definition 1.4.14 (Generation of σ-algebras). Let F be any family of sets in X. We define F to be the intersection of all the σ-algebras that con- tain F, which is again a σ-algebra by Exercise 1.4.13. Equivalently, F is the coarsest σ-algebra that contains F. We say that F is the σ-algebra generated by F. Since every σ-algebra is a Boolean algebra, we have the trivial inclusion Fbool F. However, equality need not hold it only holds if and only if Fbool is a σ- algebra. For instance, if F is the collection of all boxes in Rd, then Fbool is the elementary algebra (Exercise 1.4.7), but F cannot equal this algebra, as it is not a σ-algebra.
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