1.7. Outer measures, pre-measures, product measures 151 and μ∗(A01 ∪ A10 ∪ A11) = μ∗(A01) + μ∗(A10 ∪ A11) while from the Carath´ eodory measurability of F one has μ∗(A00 ∪ A01) = μ∗(A00) + μ∗(A01) putting these identities together we obtain (1.32). (Note that no subtraction is employed here, and so the arguments still work when some sets have infinite outer measure.) Now we verify that B is a σ-algebra. As it is already a Boolean algebra, it suﬃces (see Exercise 1.7.3 below) to verify that B is closed with respect to countable disjoint unions. Thus, let E1,E2,... be a disjoint sequence of Carath´ eodory-measurable sets, and let A be arbitrary. We wish to show that μ∗(A) = μ∗(A ∩ ∞ n=1 En) + μ∗(A\ ∞ n=1 En). In view of subadditivity, it suﬃces to show that μ∗(A) ≥ μ∗(A ∩ ∞ n=1 En) + μ∗(A\ ∞ n=1 En). For any N ≥ 1, N n=1 En is Carath´ eodory measurable (as B is a Boolean algebra), and so μ∗(A) ≥ μ∗(A ∩ N n=1 En) + μ∗(A\ N n=1 En). By monotonicity, μ∗(A\ N n=1 En) ≥ μ∗(A\ ∞ n=1 En). Taking limits as N → ∞, it thus suﬃces to show that μ∗(A ∩ ∞ n=1 En) ≤ lim N→∞ μ∗(A ∩ N n=1 En). But by the Carath´ eodory measurability of N n=1 En, we have μ∗(A ∩ N+1 n=1 En) = μ∗(A ∩ N n=1 En) + μ∗(A ∩ EN+1\ N n=1 En) for any N ≥ 0, and thus on iteration lim N→∞ μ∗(A ∩ N n=1 En) = ∞ N=0 μ∗(A ∩ EN+1\ N n=1 En).

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