62 1. Measure theory Remark 1.3.22. At least as far as Rd is concerned, an equivalent definition of local uniform convergence is: fn converges locally uniformly to f if, for every point x0 Rd, there exists an open neighbourhood U of x0 such that fn converges uniformly to f on U. The equivalence of the two definitions is immediate from the Heine-Borel theorem. More generally, the adverb “locally” in mathematics is usually used in this fashion a property P is said to hold locally on some domain X if, for every point x0 in that domain, there is an open neighbourhood of x0 in X on which P holds. One should caution, though, that on domains on which the Heine-Borel theorem does not hold, the bounded-set notion of local uniform conver- gence is not equivalent to the open-set notion of local uniform convergence (though, for locally compact spaces, one can recover equivalence if one re- places “bounded” by “compact”). Example 1.3.23. The functions x x/n on R for n = 1, 2,... converge locally uniformly (and hence pointwise) to zero on R, but do not converge uniformly. Example 1.3.24. The partial sums ∑N n=0 xn n! of the Taylor series ex = ∑∞ n=0 xn n! converges to ex locally uniformly (and hence pointwise) on R, but not uniformly. Example 1.3.25. The functions fn(x) := 1 nx 1x0 for n = 1, 2,... (with the convention that fn(0) = 0) converge pointwise everywhere to zero, but do not converge locally uniformly. From the preceding example, we see that pointwise convergence (either everywhere or almost everywhere) is a weaker concept than local uniform convergence. Nevertheless, a remarkable theorem of Egorov, which demon- strates Littlewood’s third principle, asserts that one can recover local uni- form convergence as long as one is willing to delete a set of small measure: Theorem 1.3.26 (Egorov’s theorem). Let fn : Rd C be a sequence of measurable functions that converge pointwise almost everywhere to another function f : Rd C, and let ε 0. Then there exists a Lebesgue measurable set A of measure at most ε, such that fn converges locally uniformly to f outside of A. Note that Example 1.3.25 demonstrates that the exceptional set A in Egorov’s theorem cannot be taken to have zero measure, at least if one uses the bounded-set definition of local uniform convergence from Definition 1.3.21. (If one instead takes the “open neighbourhood” definition, then the sequence in Example 1.3.25 does converge locally uniformly on R\{0} in the open neighbourhood sense, even if it does not do so in the bounded-set sense.
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