1.4. Abstract measure spaces 89 Let 0 ε 1 be arbitrary. Then we have f(x) = sup n fn(x) (1 − ε)ci for all x ∈ Ai. Thus, if we define the sets Ai,n := {x ∈ Ai : fn(x) (1 − ε)ci} then the Ai,n increase to Ai and are measurable. By upwards monotonicity of measure, we conclude that lim n→∞ μ(Ai,n) = μ(Ai). On the other hand, observe the pointwise bound fn ≥ k i=1 (1 − ε)ci1Ai,n for any n integrating this, we obtain X fn dμ ≥ (1 − ε) k i=1 ciμ(Ai,n). Taking limits as n → ∞, we obtain lim n→∞ X fn dμ ≥ (1 − ε) k i=1 ciμ(Ai) sending ε → 0 we then obtain the claim. Remark 1.4.44. It is easy to see that the result still holds if the mono- tonicity fn ≤ fn+1 only holds almost everywhere rather than everywhere. This has a number of important corollaries. First, we can generalise (part of) Tonelli’s theorem for exchanging sums (see Theorem 0.0.2): Corollary 1.4.45 (Tonelli’s theorem for sums and integrals). Let (X, B,μ) be a measure space, and let f1,f2,... : X → [0, +∞] be a sequence of un- signed measurable functions. Then one has X ∞ n=1 fn dμ = ∞ n=1 X fn dμ. Proof. Apply the monotone convergence theorem (Theorem 1.4.43) to the partial sums FN := ∑N n=1 fn. Exercise 1.4.42. Give an example to show that this corollary can fail if the fn are assumed to be absolutely integrable rather than unsigned measurable, even if the sum ∑∞ n=1 fn(x) is absolutely convergent for each x. (Hint: Think about the three escapes to infinity.)
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