1.4. Abstract measure spaces 73 (Hint: To show that two families F, F of sets generate the same σ-algebra, it suﬃces to show that every σ-algebra that contains F, contains F also, and conversely.) There is an analogue of Exercise 1.4.9, which illustrates the extent to which a generated σ-algebra is “larger” than the analogous generated Boolean algebra: Exercise 1.4.15 (Recursive description of a generated σ-algebra). (This exercise requires familiarity with the theory of ordinals, which is reviewed in §2.4 of An epsilon of room, Vol. I. Recall that we are assuming the axiom of choice throughout this text.) Let F be a collection of sets in a set X, and let ω1 be the first uncountable ordinal. Define the sets Fα for every countable ordinal α ∈ ω1 via transfinite induction as follows: (i) Fα := F. (ii) For each countable successor ordinal α = β + 1, we define Fα to be the collection of all sets that either the union of an at most countable number of sets in Fn−1 (including the empty union ∅), or the complement of such a union. (iii) For each countable limit ordinal α = supβα β, we define Fα := βα Fβ. Show that F = α∈ω1 Fα. Remark 1.4.17. The first uncountable ordinal ω1 will make several further cameo appearances here and in An epsilon of room, Vol. I, for instance, by generating counterexamples to various plausible statements in point-set topology. In the case when F is the collection of open sets in a topological space, so that F, then the sets Fα are essentially the Borel hierarchy (which starts at the open and closed sets, then moves on to the Fσ and Gδ sets, and so forth) these play an important role in descriptive set theory. Exercise 1.4.16. (This exercise requires familiarity with the theory of car- dinals.) Let F be an infinite family of subsets of X of cardinality κ (thus κ is an infinite cardinal). Show that F has cardinality at most κℵ0 . (Hint: Use Exercise 1.4.15.) In particular, show that the Borel σ-algebra B[Rd] has cardinality at most c := 2ℵ0 . Conclude that there exist Jordan measurable (and hence Lebesgue mea- surable) subsets of Rd which are not Borel measurable. (Hint: How many subsets of the Cantor set are there?) Use this to place the Borel σ-algebra on the diagram that you drew for Exercise 1.4.8. Remark 1.4.18. Despite this demonstration that not all Lebesgue mea- surable subsets are Borel measurable, it is remarkably diﬃcult (though not

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