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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.