1.3. The Lebesgue integral 63 On a domain such as Rd\A, bounded-set locally uniform convergence implies open-neighbourhood locally uniform convergence, but not conversely.) Proof. By modifying fn and f on a set of measure zero (that can be ab- sorbed into A at the end of the argument) we may assume that fn converges pointwise everywhere to f, thus for every x ∈ Rd and m 0 there exists N ≥ 0 such that |fn(x) − f(x)| ≤ 1/m for all n ≥ N. We can rewrite this fact set-theoretically as ∞ N=0 EN,m = ∅ for each m, where EN,m := {x ∈ Rd : |fn(x) − f(x)| 1/m for some n ≥ N}. It is clear that the EN,m are Lebesgue measurable, and are decreasing in N. Applying downward monotone convergence (Exercise 1.2.11(ii)) we conclude that, for any radius R 0, one has lim N→∞ m(EN,m ∩ B(0,R)) = 0. (The restriction to the ball B(0,R) is necessary, because the downward monotone convergence property only works when the sets involved have finite measure.) In particular, for any m ≥ 1, we can find Nm such that m(EN,m ∩ B(0,m)) ≤ ε 2m for all N ≥ Nm. Now let A := ∞ m=1 ENm,m ∩ B(0,m). Then A is Lebesgue measurable, and by countable subadditivity, m(A) ≤ ε. By construction, we have |fn(x) − f(x)| ≤ 1/m whenever m ≥ 1, x ∈ Rd\A, |x| ≤ m, and n ≥ Nm. In particular, we see for any ball B(0,m0) with an integer radius, fn converges uniformly to f on B(0,m0)\A. Since every bounded set is contained in such a ball, the claim follows. Remark 1.3.27. Unfortunately, one cannot in general upgrade local uni- form convergence to uniform convergence in Egorov’s theorem. A basic example here is the moving bump example, fn := 1[n,n+1] on R, which “es- capes to horizontal infinity”. This sequence converges pointwise (and locally uniformly) to the zero function f ≡ 0. However, for any 0 ε 1 and any n, we have |fn(x) − f(x)| ε on a set of measure 1, namely on the interval [n, n + 1]. Thus, if one wanted fn to converge uniformly to f outside of a set A, then that set A has to contain a set of measure 1. In fact, it must contain the intervals [n, n + 1] for all sufficiently large n and must therefore have infinite measure.
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.