1.7. Outer measures, pre-measures, product measures 171 By combining Fubini’s theorem with Tonelli’s theorem, we can recast the absolute integrability hypothesis: Corollary 1.7.23 (Fubini-Tonelli theorem). Let (X, BX , μX) and (Y, BY , μY ) be complete σ-finite measure spaces, and let f : X × Y → C be measurable with respect to BX × BY . If X ( Y |f(x, y)| dμY (y)) dμX(x) ∞ (note the left-hand side always exists, by Tonelli’s theorem), then f is abso- lutely integrable with respect to BX × BY , and in particular, the conclusions of Fubini’s theorem hold. Similarly, if we use Y ( X |f(x, y)| dμX(x)) dμY (y) instead of X ( Y |f(x, y)| dμY ) dμX. The Fubini-Tonelli theorem is an indispensable tool for computing inte- grals. We give some basic examples below: Exercise 1.7.24 (Area interpretation of integral). Let (X, B,μ) be a σ- finite measure space, and let R be equipped with Lebesgue measure m and the Borel σ-algebra B[R]. Show that if f : X → [0, +∞] is measurable if and only if the set {(x, t) ∈ X × R : 0 ≤ t ≤ f(x)} is measurable in B × B[R], in which case we have (μ × m)({(x, t) ∈ X × R : 0 ≤ t ≤ f(x)}) = X f(x) dμ(x). Similarly, if we replace {(x, t) ∈ X × R : 0 ≤ t ≤ f(x)} by {(x, t) ∈ X × R : 0 ≤ t f(x)}. Exercise 1.7.25 (Distribution formula). Let (X, B,μ) be a σ-finite measure space, and let f : X → [0, +∞] be measurable. Show that X f(x) dμ(x) = [0,+∞] μ({x ∈ X : f(x) ≥ λ}) dλ. (Note that the integrand on the right-hand side is monotone and thus Lebesgue measurable.) Show that this identity continues to hold if we re- place {x ∈ X : f(x) ≥ λ} by {x ∈ X : f(x) λ}. Exercise 1.7.26 (Approximations to the identity). Let P : Rd → R+ be a good kernel (see Exercise 1.6.27), and let Pt(x) := 1 td P ( x t ) be the associated rescaled functions. Show that if f : Rd → C is absolutely integrable, that f ∗Pt converges in L1 norm to f as t → 0. (Hint: Use the density argument. You will need an upper bound on f ∗Pt L1(Rd) which can be obtained using Tonelli’s theorem.)

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