1.4. Abstract measure spaces 87 To put it another way: When can we ensure that one can interchange inte- grals and limits, lim n→∞ X fn ? = X lim n→∞ fn dμ? There are certainly some cases in which one can safely do this: Exercise 1.4.41 (Uniform convergence on a finite measure space). Suppose that (X, B,μ) is a finite measure space (so μ(X) ∞), and fn : X [0, +∞] (resp. fn : X C) are a sequence of unsigned measurable functions (resp. absolutely integrable functions) that converge uniformly to a limit f. Show that X fn converges to X f dμ. However, there are also cases in which one cannot interchange limits and integrals, even when the fn are unsigned. We give the three classic examples, all of “moving bump” type, though the way in which the bump moves varies from example to example: Example 1.4.39 (Escape to horizontal infinity). Let X be the real line with Lebesgue measure, and let fn := 1[n,n+1]. Then fn converges pointwise to f := 0, but R fn(x) dx = 1 does not converge to R f(x) dx = 0. Somehow, all the mass in the fn has escaped by moving off to infinity in a horizontal direction, leaving none behind for the pointwise limit f. Example 1.4.40 (Escape to width infinity). Let X be the real line with Lebesgue measure, and let fn := 1 n 1[0,n]. Then fn now converges uniformly to f := 0, but R fn(x) dx = 1 still does not converge to R f(x) dx = 0. Exercise 1.4.41 would prevent this from happening if all the fn were supported in a single set of finite measure, but the increasingly wide nature of the support of the fn prevents this from happening. Example 1.4.41 (Escape to vertical infinity). Let X be the unit interval [0, 1] with Lebesgue measure (restricted from R), and let fn := n1[ 1 n , 2 n ] . Now, we have finite measure, and fn converges pointwise to f, but no uniform convergence. And again, [0,1] fn(x) dx = 1 is not converging to [0,1] f(x) dx = 0. This time, the mass has escaped vertically, through the increasingly large values of fn. Remark 1.4.42. From the perspective of time-frequency analysis (or per- haps more accurately, space-frequency analysis), these three escapes are analogous (though not quite identical) to escape to spatial infinity, escape to zero frequency, and escape to infinite frequency respectively, thus describ- ing the three different ways in which phase space fails to be compact (if one excises the zero frequency as being singular).
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