114 1. Measure theory Next, we observe the quantitative estimate (1.22) Rd |fh(x) f(x)| dx 2 Rd |f(x)| dx for any h Rd. This follows easily from the triangle inequality Rd |fh(x) f(x)| dx Rd |fh(x)| dx + Rd |f(x)| dx together with the translation invariance of the Lebesgue integral: Rd |fh(x)| dx = Rd |f(x)| dx. Now we put the two ingredients together. Let f : Rd C be absolutely integrable, and let ε 0 be arbitrary. Applying Littlewood’s second prin- ciple (Theorem 1.3.20(iii)) to the absolutely integrable function F , we can find a continuous, compactly supported function g : Rd C such that Rd |f(x) g(x)| dx ε. Applying (1.22), we conclude that Rd |(f g)h(x) (f g)(x)| dx 2ε, which we rearrange as Rd |(fh f)h(x) (gh g)(x)| dx 2ε. By the dense subclass result, we also know that Rd |gh(x) g(x)| dx ε for all h sufficiently close to zero. From the triangle inequality, we conclude that Rd |fh(x) f(x)| dx for all h sufficiently close to zero, and the claim follows. Remark 1.6.14. In the above application of the density argument, we proved the required quantitative estimate directly for all functions f in the original class of functions. However, it is also possible to use the density argument a second time and initially verify the quantitative estimate just for functions f in a nice subclass (e.g. continuous functions of compact support). In many cases, one can then extend that estimate to the general case by using tools such as Fatou’s lemma (Corollary 1.4.46), which are particularly suited for showing that upper bound estimates are preserved with respect to limits.
Previous Page Next Page