166 1. Measure theory Next, let D be the set of all E ∈ B such that F \E, E\F , F ∩ E, and X\(E ∪ F ) all lie in B for all F ∈ B. By the previous discussion, we see that D contains A. One also easily verifies that D is a monotone class. By definition of B, we conclude that D = B. Since B is also closed under complements, this implies that B is closed with respect to finite unions. Since this class also contains A, which contains ∅, we conclude that B is a Boolean algebra. Since B is also closed under increasing countable unions, we conclude that it is closed under arbitrary countable unions, and is thus a σ-algebra. As it contains A, it must also contain A. Theorem 1.7.15 (Tonelli’s theorem, incomplete version). Let (X, BX,μX) and (Y, BY , μY ) be σ-finite measure spaces, and let f : X × Y → [0, +∞] be measurable with respect to BX × BY . Then: (i) The functions x → Y f(x, y) dμY (y) and y → X f(x, y) dμX(x) (which are well defined, thanks to Exercise 1.7.18) are measurable with respect to BX and BY , respectively. (ii) We have X×Y f(x, y) dμX × μY (x, y) = X ( Y f(x, y) dμY (y)) dμX(x) = Y ( X f(x, y) dμX(x)) dμY (y). Proof. By writing the σ-finite space X as an increasing union X = ∞ n=1 Xn of finite measure sets, we see from several applications of the monotone convergence theorem (Theorem 1.4.43) that it suﬃces to prove the claims with X replaced by Xn. Thus we may assume without loss of generality that X has finite measure. Similarly we may assume Y has finite measure. Note from (1.36) that this implies that X × Y has finite measure also. Every unsigned measurable function is the increasing limit of unsigned simple functions. By several applications of the monotone convergence the- orem (Theorem 1.4.43), we thus see that it suﬃces to verify the claim when f is a simple function. By linearity, it then suﬃces to verify the claim when f is an indicator function, thus f = 1S for some S ∈ BX × BY . Let C be the set of all S ∈ BX × BY for which the claims hold. From the repeated applications of the monotone convergence theorem (Theorem 1.4.43) and the downward monotone convergence theorem (which is available in this finite measure setting) we see that C is a monotone class. By direct computation (using (1.36)), we see that C contains as an el- ement any product S = E × F with E ∈ BX and F ∈ BY . By finite

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