1.4. Abstract measure spaces 91 Remark 1.4.47. Informally, Fatou’s lemma tells us that when taking the pointwise limit of unsigned functions fn, that mass X fn dμ can be de- stroyed in the limit (as was the case in the three key moving bump examples), but it cannot be created in the limit. Of course the unsigned hypothesis is necessary here (consider for instance multiplying any of the moving bump examples by −1). While this lemma was stated only for pointwise limits, the same general principle (that mass can be destroyed, but not created, by the process of taking limits) tends to hold for other “weak” notions of convergence. See §1.9 of An epsilon of room, Vol. I, for some examples of this. Finally, we give the other major way to shut down loss of mass via escape to infinity, which is to dominate all of the functions involved by an absolutely convergent one. This result is known as the dominated convergence theorem: Theorem 1.4.48 (Dominated convergence theorem). Let (X, B,μ) be a measure space, and let f1,f2,... : X → C be a sequence of measurable functions that converge pointwise μ-almost everywhere to a measurable limit f : X → C. Suppose that there is an unsigned absolutely integrable function G: X → [0, +∞] such that |fn| are pointwise μ-almost everywhere bounded by G for each n. Then we have lim n→∞ X fn dμ = X f dμ. From the moving bump examples we see that this statement fails if there is no absolutely integrable dominating function G. The reader is encouraged to see why, in each of the moving bump examples, no such dominating function exists, without appealing to the above theorem. Note also that when each of the fn is an indicator function fn = 1En , the dominated convergence theorem collapses to Exercise 1.4.24. Proof. By modifying fn,f on a null set, we may assume without loss of generality that the fn converge to f pointwise everywhere rather than μ- almost everywhere, and similarly we can assume that |fn are bounded by G pointwise everywhere rather than μ-almost everywhere. By taking real and imaginary parts we may assume without loss of gen- erality that fn,f are real, thus −G ≤ fn ≤ G pointwise. Of course, this implies that −G ≤ f ≤ G is pointwise also. If we apply Fatou’s lemma (Corollary 1.4.46) to the unsigned functions fn + G, we see that X f + G dμ ≤ lim inf n→∞ X fn + G dμ,

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