90 1. Measure theory Exercise 1.4.43 (Borel-Cantelli lemma). Let (X, B,μ) be a measure space, and let E1,E2,E3,... be a sequence of B-measurable sets such that ∞ n=1 μ(En) ∞. Show that almost every x ∈ X is contained in at most finitely many of the En (i.e. {n ∈ N : x ∈ En} is finite for almost every x ∈ X). (Hint: Apply Tonelli’s theorem to the indicator functions 1En .) Exercise 1.4.44. (i) Give an alternate proof of the Borel-Cantelli lemma (Exercise 1.4.43) that does not go through any of the convergence theorems, but in- stead exploits the more basic properties of measure from Exercise 1.4.23. (ii) Give a counterexample that shows that the Borel-Cantelli lemma can fail if the condition ∑∞ n=1 μ(En) ∞ is relaxed to limn→∞ μ(En) = 0. Second, when one does not have monotonicity, one can at least obtain an important inequality, known as Fatou’s lemma: Corollary 1.4.46 (Fatou’s lemma). Let (X, B,μ) be a measure space, and let f1,f2,... : X → [0, +∞] be a sequence of unsigned measurable functions. Then X lim inf n→∞ fn dμ ≤ lim inf n→∞ X fn dμ. Proof. Write FN := infn≥N fn for each N. Then the FN are measurable and non-decreasing, and hence by the monotone convergence theorem (Theorem 1.4.43) X sup N0 FN dμ = sup N0 X FN dμ. By definition of lim inf, we have supN0 FN = lim infn→∞ fn. By mono- tonicity, we have X FN dμ ≤ X fn dμ for all n ≥ N, and thus X FN dμ ≤ inf n≥N X fn dμ. Hence we have X lim inf n→∞ fn dμ ≤ sup N0 inf n≥N X fn dμ. The claim then follows by another appeal to the definition of the lim inferior.

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