MATH 6280 MEASURE THEORY — NOTES ON OUTER MEASURES Definition: Let X be a set. The power set of X, written P(X), is the collection of all subsets of X. Definition: Let S be a collection of subsets of X. We say that a set function ω : S → [0, ∞] is monotone if ω(E) ≤ ω(F ) whenever E and F are members of S such that E ⊂ F . ∞ We say that ω is countably monotone if ω(E) ≤ i=1 ω(Ei) whenever E, E1, E2,... ∞ are members of S such that E ⊂ ∪ =1Ei. i P Definition: An outer measure on X is a set function µ∗ : P(X) → [0, ∞] such that (i) µ∗(∅) = 0, (ii) µ∗ is monotone, and (iii) µ∗ is countably monotone. Note: In the above definition, (ii) is implied by (i) and (iii). Definition: Let µ∗ be an outer measure on X. A set E ⊂ X is µ∗-measurable if µ∗(A) = µ∗(A ∩ E) + µ∗(A ∩ Ec) for every A ⊂ X. We write M∗ to denote the collection of all µ∗-measurable sets. Theorem 17.8 [12.1]: Let µ∗ be an outer measure. Then: (a) M∗ is a σ-algebra. (b) Let µ¯ be the restriction of µ∗ to M∗. Then µ¯ is a measure on M∗. (c) µ¯ is complete. Definition: Let S be a collection of subsets of X. Let ω : S → [0, ∞] be a set function. Define the set function ω∗ : P(X ) → [0, ∞] as follows: ω∗(∅) = 0,and ∗ ∞ ∞ ω (E) = inf { i=1 : E1, E2,... ∈ S and E ⊂ ∪i=1Ei} for every E ⊂ X such that E = ∅. (We define theP above inf to be +∞ if there exists no countable collection E1, E2,... ∈ S whose union contains E.) 1 Theorem 17.9 [Lemma 12.4]: The set function ω∗ as defined above is an outer measure. We call ω∗ the outer measure induced by ω. Definition: Given a set function ω : S → [0, ∞], • let ω∗ be the outer measure induced by ω, • let M∗ be the class of ω∗-measurable sets, and • letω ¯ be the restriction of ω∗ to M∗. Thenω ¯ is called the Carath´eodory measure induced by ω. Notation: Let S be a collection of subsets of X. We write Sσ to denote the class of sets that are countable unions of members of S, and we write Sσδ to denote the class of sets that are countable intersections of members of Sσ. Thus, S ⊂ Sσ ⊂ Sσδ ⊂ σ[S]. Proposition: For ω etc. as above, assume E ⊂ X and ω∗(E) < ∞. Then: ∗ ∗ (i) There exists A ∈ Sσδ such that E ⊂ A and ω (A) = ω (E). (ii) If E ∈M∗ and S⊂M∗, then A ∈M∗ and ω¯(A ∼ E)=0. Terminology: Consider the Carath´eodory measureω ¯ : M∗ → [0, ∞] in- duced by ω : S → [0, ∞]. We say thatω ¯ is an extension of ω if S⊂M∗ and ω¯(E) = ω(E) for every E in S. Definition: A set function ω : S → [0, ∞] is a premeasure if the following all hold: (a) If ∅ ∈ S, then ω(∅) = 0; n n (b) If E1,...,En are disjoint members of S, and if ∪i=1Ei ∈ S, then ω(∪i=1Ei) = n i=1 ω(Ei) (“finite additivity”); (c) ω is countably monotone. P Definition: A collection S is closed under relative complements if A ∼ B ∈ S whenever A and B are both members of S. Notes: If S is closed under relative complements, then (i) ∅ ∈ S (unless S is empty), (ii) S is closed under finite intersections, and (iii) If X ∈ S, then S is closed under unions. Thus, if X ∈ S and S is closed under relative complements, then S is an algebra. 2 Moreover, every premeasure on an algebra S is a measure on the algebra ∞ (i.e., for disjoint E1, E2,... ∈ S such that ∪i=1Ei ∈ S, we have ω(∪Ei) = ω(Ei)). TerminologyP : For convenience, we will refer to the three assumptions • ω is a premeasure on S, • S is nonempty, and • S is closed under relative complements together as Condition C. Clearly Condition C holds if S is an algebra and ω is a measure. (Indeed, if X ∈ S, then this is equivalent to condition C.) Theorem 17.12 [12.3,12.5] Assume that Condition C holds. Then the Carath´eodory measure ω¯ induced by ω is an extension of ω. In the situation of the above result, we callω ¯ the Carath´eodory extension of ω. Definition: Let S be a nonempty collection of subsets of X. We say that S is a semiring if the following hold: • if A, B ∈ S, then A ∩ B ∈ S; and • if A, B ∈ S, then there exist a finite disjoint collection C1,...,Cn n of sets in S such that A ∼ B = ∪i=1Ci. (If S is a semiring and X ∈ S, then S is called a semi-algebra.) Property: Let S be a semiring, and let S∪ be the collection of all unions of finite disjoint collections of sets in S. Then S∪ is closed under relative complements. Note: If S is a semiring and X ∈ S∪, then S∪ is an algebra, and hence S∪ = A[S]. 3 Theorem 17.13 [12.9] Let ω be a premeasure on a semiring S. Then ω has a unique extension to a premeasure on S∪. Note: If also X ∈ S, then the above conclusion says that ω has a unique extension to a measure on A[S]; this is how [12.9] is stated. Proof: Clearly there is at most one possible way to extend a premeasure ω on S to a premeasure ω∪ on S∪, since ω∪ must be finitely additive; namely, n n ∪ If E = ∪i=1Ei (with E1,...,En ∈ S disjoint), then ω (E) = ω(Ei). i=1 X We must show that the above ω∪ is well-defined; that is, if we also have m E = ∪j=1Fj for some other disjoint collection of sets F1,...,Fm ∈ S, then ∪ m we must also have ω (E) = j=1 ω(Fj). The key to this is the fact that each Fj ∩ Ei is in S (this is a semiring property). Hence P m m n n m n ω(Fj) = ω(Fj ∩ Ei) = ω(Fj ∩ Ei) = ω(Ei). j=1 j=1 i=1 i=1 j=1 i=1 X X X X X X Therefore ω∪ is well-defined. Now we must show that ω∪ is a premeasure. Finite additivity is clear, so it only remains to show that ∞ ∞ ∪ ∪ ∪ If E ⊂ Ek with E, E1, E2,... ∈ S , then ω (E) ≤ ω (Ek). k=1 k=1 [ X n Assume E, E1, E2,... are as above. Write E = ∪i=1Ci with C1,...,Cn ∞ ∞ disjoint members of S. We can write ∪k=1Ek = ∪k=1Dk where D1,D2,... are ∪ disjoint members of S (let D1 = E1, D2 = E2 ∼ E1, D3 = (E3 ∼ E2) ∼ ∪ ∞ ∞ E1, and so on). Next, by the definition of S , we can write ∪k=1Dk = ∪l=1Gl where G1,G2,... are disjoint members of S (note that we require S here, no S∪). Then we have ∞ ∞ ∞ ∪ ∪ ω(Gl) = ω (Dk) ≤ ω (Ek) (1) l=1 k=1 k=1 X X X (the equality comes from the definition of ω∪ in terms of ω; and we get the 4 inequality because Dk ⊂ Ek for each k). Finally, n ∪ ω (E) = ω(Ci) i=1 Xn ∞ ∞ ≤ ω(Ci ∩ Gl) (because Ci = ∪l=1(Ci ∩ Gl); i=1 l=1 ! X X note that Ci ∩ Gl ∈ S ) ∞ n = ω(Ci ∩ Gl) l=1 i=1 X∞ X = ω(E ∩ Gl) (because ω is finitely additive on S l=1 X n and ∪i=1(Ci ∩ Gl) = E ∩ Gl ) ∞ ≤ ω(Gl) (because ω is monotone on S) l=1 X∞ ∪ ≤ ω (Ek) (by Equation (1) above). k=1 X This concludes the proof. Definition: Let ω be a premeasure on S. We say that ω is σ-finite if there ∞ exist X1,X2,... ∈ S such that X = ∪k=1 and ω(Xk) < ∞ for every k. (Note that if S is closed under complements, then we can choose X1,X2,... to be disjoint; see the proof of Theorem 17.13 [12.9] above.) Carath´eodory(-Hahn) Extension Theorem (p. 356) [Theorem 12.8 (and 12.9)] Let ω : S → [0, ∞] be a premeasure on a semiring S. [This holds if ω is a measure on an algebra S.] Then (a) The Carath´eodory measure induced by ω, ω¯, is an extension of ω; and (b) If ω is σ-finite, then ω¯ is also σ-finite and ω¯ is the only measure on M∗ (or on σ[S]) that extends ω. 5.