92 1. Measure theory which on subtracting the finite quantity X G gives X f lim inf n→∞ X fn dμ. Similarly, if we apply that lemma to the unsigned functions G fn, we obtain X G f lim inf n→∞ X G fn negating this inequality and then cancelling X G again we conclude that lim sup n→∞ X fn X f dμ. The claim then follows by combining these inequalities. Remark 1.4.49. We deduced the dominated convergence theorem from Fatou’s lemma, and Fatou’s lemma from the monotone convergence theorem. However, one can obtain these theorems in a different order, depending on one’s taste, as they are so closely related. For instance, in [StSk2005], the logic is somewhat different one first obtains the slightly simpler bounded convergence theorem, which is the dominated convergence theorem under the assumption that the functions are uniformly bounded and all supported on a single set of finite measure, and then uses that to deduce Fatou’s lemma, which in turn is used to deduce the monotone convergence theorem and then the horizontal and vertical truncation properties are used to extend the bounded convergence theorem to the dominated convergence theorem. It is instructive to view a couple different derivations of these key theorems to get more of an intuitive understanding as to how they work. Exercise 1.4.45. Under the hypotheses of the dominated convergence the- orem (Theorem 1.4.48), establish also that fn f L1 0 as n ∞. Exercise 1.4.46 (Almost dominated convergence). Let (X, B,μ) be a mea- sure space, and let f1,f2,... : X C be a sequence of measurable func- tions that converge pointwise μ-almost everywhere to a measurable limit f : X C. Suppose that there is an unsigned absolutely integrable func- tion G, g1,g2,... : X [0, +∞] such that the |fn| are pointwise μ-almost everywhere bounded by G + gn, and that X gn 0 as n ∞. Show that lim n→∞ X fn = X f dμ. Exercise 1.4.47 (Defect version of Fatou’s lemma). Let (X, B,μ) be a measure space, and let f1,f2,... : X [0, +∞] be a sequence of unsigned absolutely integrable functions that converges pointwise to an absolutely
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