152 1. Measure theory On the other hand, from countable subadditivity one has μ∗(A ∩ ∞ n=1 En) ≤ ∞ N=0 μ∗(A ∩ EN+1\ N n=1 En) and the claim follows. Finally, we show that μ is a measure. It is clear that μ(∅) = 0, so it suﬃces to establish countable additivity, thus we need to show that μ∗( ∞ n=1 En) = ∞ n=1 μ∗(En) whenever E1,E2,... are Carath´ eodory-measurable and disjoint. By subad- ditivity it suﬃces to show that μ∗( ∞ n=1 En) ≥ ∞ n=1 μ∗(En). By monotonicity it suﬃces to show that μ∗( N n=1 En) = N n=1 μ∗(En) for any finite N. But from the Carath´ eodory measurability of N n=1 En one has μ∗( N+1 n=1 En) = μ∗( N n=1 En) + μ∗(EN+1) for any N ≥ 0, and the claim follows from induction. Exercise 1.7.3. Let B be a Boolean algebra on a set X. Show that B is a σ-algebra if and only if it is closed under countable disjoint unions, which means that ∞ n=1 En ∈ B whenever E1,E2,E3,... ∈ B are a countable sequence of disjoint sets in B. Remark 1.7.4. Note that the above theorem, combined with Exercise 1.7.2 gives a slightly alternate way to construct Lebesgue measure from Lebesgue outer measure than the construction given in Section 1.2. This is arguably a more eﬃcient way to proceed, but is also less geometrically intuitive than the approach taken in Section 1.2. Remark 1.7.5. From Exercise 1.7.1 we see that the measure μ constructed by the Carath´ eodory extension theorem is automatically complete (see Def- inition 1.4.31). Remark 1.7.6. In §1.15 of An epsilon of room, Vol. I, an important exam- ple of a measure constructed by Carath´ eodory’s theorem is given, namely the d-dimensional Hausdorff measure Hd on Rn that is good for measuring the size of d-dimensional subsets of Rn.

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