112 1. Measure theory (ii) A number of differentiation theorems, which assert, for instance, that monotone, Lipschitz, or bounded variation functions in one dimension are almost everywhere differentiable. (iii) The second fundamental theorem of calculus for absolutely contin- uous functions. 1.6.1. The Lebesgue differentiation theorem in one dimension. The main objective of this section is to show Theorem 1.6.11 (Lebesgue differentiation theorem, one-dimensional case). Let f : R C be an absolutely integrable function, and let F : R C be the definite integral F (x) := [−∞,x] f(t) dt. Then F is continuous and almost everywhere differentiable, and F (x) = f(x) for almost every x R. This can be viewed as a variant of Corollary 1.6.10 the hypotheses are weaker because f is only assumed to be absolutely integrable, rather than continuous (and can live on the entire real line, and not just on a compact interval) but the conclusion is weaker too, because F is only found to be al- most everywhere differentiable, rather than everywhere differentiable. (But such a relaxation of the conclusion is necessary at this level of generality consider for instance the example when f = 1[0,1].) The continuity is an easy exercise: Exercise 1.6.5. Let f : R C be an absolutely integrable function, and let F : R C be the definite integral F (x) := [−∞,x] f(t) dt. Show that F is continuous. The main difficulty is to show that F (x) = f(x) for almost every x R. This will follow from Theorem 1.6.12 (Lebesgue differentiation theorem, second formulation). Let f : R C be an absolutely integrable function. Then (1.20) lim h→0+ 1 h [x,x+h] f(t) dt = f(x) for almost every x R, and (1.21) lim h→0+ 1 h [x−h,x] f(t) dt = f(x) for almost every x R. Exercise 1.6.6. Show that Theorem 1.6.11 follows from Theorem 1.6.12. We will just prove the first fact (1.20) the second fact (1.21) is similar (or can be deduced from (1.20) by replacing f with the reflected function x f(−x).
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