102 1. Measure theory Corollary 1.5.10. Suppose that fn : X → C are a sequence of measurable functions that converge in L1 norm to a limit f. Then there exists a sub- sequence fnj that converges almost uniformly (and hence, pointwise almost everywhere) to f (while remaining convergent in L1 norm, of course). Proof. Since fn −f L1(μ) → 0 as n → ∞, we can select n1 n2 n3 . . . such that fnj −f L1(μ) ≤ 2−j (say). This is enough for the previous exercise to apply. Actually, one can strengthen this corollary a bit by relaxing L1 conver- gence to convergence in measure: Exercise 1.5.6. Suppose that fn : X → C are a sequence of measur- able functions that converge in measure to a limit f. Then there ex- ists a subsequence fnj that converges almost uniformly (and hence, point- wise almost everywhere) to f. (Hint: Choose the nj so that the sets {x ∈ X : |fnj (x) − f(x)| 1/j} have a suitably small measure.) It is instructive to see how this subsequence is extracted in the case of the typewriter sequence. In general, one can view the operation of passing to a subsequence as being able to eliminate “typewriter” situations in which the tail support is much larger than the width. Exercise 1.5.7. Let (X, B,μ) be a measure space, let fn : X → C be a sequence of measurable functions converging pointwise almost everywhere as n → ∞ to a measurable limit f : X → C, and for each n, let fn,m : X → C be a sequence of measurable functions converging pointwise almost everywhere as m → ∞ (keeping n fixed) to fn. (i) If μ(X) is finite, show that there exists a sequence m1,m2,... such that fn,mn converges pointwise almost everywhere to f. (ii) Show the same claim is true if, instead of assuming that μ(X) is finite, we merely assume that X is σ-finite, i.e., it is the countable union of sets of finite measure. (The claim can fail if X is not σ-finite. A counterexample is if X = NN with counting measure, fn and f are identically zero for all n ∈ N, and fn,m is the indicator function of the space of all sequences (ai)i∈N ∈ NN with an ≥ m.) Exercise 1.5.8. Let fn : X → C be a sequence of measurable functions, and let f : X → C be another measurable function. Show that the following are equivalent: (i) fn converges in measure to f. (ii) Every subsequence fnj of the fn has a further subsequence fnji that converges almost uniformly to f.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.