1.4. Abstract measure spaces 85 (ii) If μ1,μ2,... are a sequence of measures on B, show that X f d ∞ n=1 μn = ∞ n=1 X f dμn. Exercise 1.4.37 (Change of variables formula). Let (X, B,μ) be a measure space, and let φ: X → Y be a measurable morphism (as defined in Remark 1.4.33) from (X, B) to another measurable space (Y, C). Define the pushfor- ward φ∗μ: C → [0, +∞] of μ by φ by the formula φ∗μ(E) := μ(φ−1(E)). (i) Show that φ∗μ is a measure on C, so that (Y, C,φ∗μ) is a measure space. (ii) If f : Y → [0, +∞] is measurable, show that Y f dφ∗μ = X (f ◦ φ) dμ. (Hint: The quickest proof here is via the monotone convergence theorem (Theorem 1.4.43) below, but it is also possible to prove the exercise without this theorem.) Exercise 1.4.38. Let T : Rd → Rd be an invertible linear transformation, and let m be Lebesgue measure on Rd. Show that T∗m = 1 | det T | m, where the pushforward T∗m of m was defined in Exercise 1.4.37. Exercise 1.4.39 (Sums as integrals). Let X be an arbitrary set (with the discrete σ-algebra), let # be counting measure (see Exercise 1.4.26), and let f : X → [0, +∞] be an arbitrary unsigned function. Show that f is measurable with X f d# = x∈X f(x). Once one has the unsigned integral, one can define the absolutely con- vergent integral exactly as in the Lebesgue case: Definition 1.4.38 (Absolutely convergent integral). Let (X, B,μ) be a mea- sure space. A measurable function f : X → C is said to be absolutely inte- grable if the unsigned integral f L1(X,B,μ) := X |f| dμ is finite, and use L1(X, B,μ), L1(X), or L1(μ) to denote the space of abso- lutely integrable functions. If f is real-valued and absolutely integrable, we define the integral X f dμ by the formula X f dμ := X f+ dμ − X f− dμ

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