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150 1. Measure theory Definition 1.7.2 (Carath´ eodory measurability). Let μ∗ be an outer mea- sure on a set X. A set E ⊂ X is said to be Carath´ eodory measurable with respect to μ∗ if one has μ∗(A) = μ∗(A ∩ E) + μ∗(A\E) for every set A ⊂ X. Exercise 1.7.1 (Null sets are Carath´ eodory measurable). Suppose that E is a null set for an outer measure μ∗ (i.e. μ∗(E) = 0). Show that E is Carath´ eodory measurable with respect to μ∗. Exercise 1.7.2 (Compatibility with Lebesgue measurability). Show that a set E ⊂ Rd is Carath´ eodory measurable with respect to Lebesgue outer measurable if and only if it is Lebesgue measurable. (Hint: One direction follows from Exercise 1.2.17. For the other direction, first verify simple cases, such as when E is a box, or when E or A are bounded.) The construction of Lebesgue measure can then be abstracted as follows: Theorem 1.7.3 (Carath´ eodory extension theorem). Let μ∗ : 2X → [0, +∞] be an outer measure on a set X, let B be the collection of all subsets of X that are Carath´ eodory measurable with respect to μ∗, and let μ: B → [0, +∞] be the restriction of μ∗ to B (thus μ(E) := μ∗(E) whenever E ∈ B). Then B is a σ-algebra, and μ is a measure. Proof. We begin with the σ-algebra property. It is easy to see that the empty set lies in B, and that the complement of a set in B lies in B also. Next, we verify that B is closed under finite unions (which will make B a Boolean algebra). Let E, F ∈ B, and let A ⊂ X be arbitrary. By definition, it suffices to show that (1.31) μ∗(A) = μ∗(A ∩ (E ∪ F )) + μ∗(A\(E ∪ F )). To simplify the notation, we partition A into the four disjoint sets A00 := A\(E ∪ F ), A10 := (A\F ) ∩ E, A01 := (A\E) ∩ F, A11 := A ∩ E ∩ F (the reader may wish to draw a Venn diagram here to understand the nature of these sets). Thus (1.31) becomes (1.32) μ∗(A00 ∪ A01 ∪ A10 ∪ A11) = μ∗(A01 ∪ A10 ∪ A11) + μ∗(A00). On the other hand, from the Carath´ eodory measurability of E, one has μ∗(A00 ∪ A01 ∪ A10 ∪ A11) = μ∗(A00 ∪ A01) + μ∗(A10 ∪ A11)