498 Chapter 4. Integration
1ifx∈{a1,...,ak} fk(x)= 4.11.10 Then 0 otherwise. Z 1 fk(x) dx = 0 for all k,4.11.11 0
but limk→∞ fk is the function that is 1 on the rationals and 0 on the irrationals between 0 and 1, and hence not integrable. 4 The pitfalls of disappearing mass can be avoided by the dominated con- vergence theorem for Riemann integrals, theorem 4.11.4. In practice this theorem is not as useful as one might hope, because the hypothesis that the limit is Riemann integrable is rarely satis ed unless the convergence is Figure 4.11.1. uniform, in which case the much easier theorem 4.11.2 applies. But the- Henri Lebesgue (1875–1941) orem 4.11.4 is the key tool in our approach to Lebesgue integration. The After Lebesgue’s father died of proof, in appendix A.21, is quite di cult and very tricky. tuberculosis, leaving three chil- dren, the oldest ve years old, Theoremin 4.11.4 (The dominated convergence theorem for Rie- his mother cleaned houses to sup- n mann integrals). Let fk : R → R be a sequence of integrable func- port them. Lebesgue later wrote, tions. Suppose that there exists R such that all f have their support in “My rst good fortune was to be k | | Rn → R born to intelligent parents, then BR, and all satisfy fk R. Let f : be an integrable function such that the set of x where lim →∞ f (x) =6 f(x) has measure 0. Then to have been sickly and extremely Z k k Z poor, which kept me from violent n n games and distractions, ... and lim fk(x) |d x| = f(x) |d x|. k→∞ most of all to have an extraordi- Rn Rn nary mother even for France, this country of good mothers.” De ning the Lebesgue integral When one of Lebesgue’s stu- dents, anxious and apprehensive, The weakness of theorem 4.11.4 is that we have to know that the limit is arrived for her rst teaching job, integrable. Usually we don’t know this; most often, we need to deal with replacing a popular substitute the limit of a sequence of functions, and all we know is that it is a limit. But we will now see that theorem 4.11.4 can be used to construct the Lebesgue teacher, she found a note from him integral, which is much better behaved under limits. waiting for her. Faites-vous aimer We abbreviate “Riemann integrable” as “R-integrable” and “Lebesgue- l a-bas comme partout, he had writ- integrable” as “L-integrable.” Proposition 4.11.5 is proved in appendix ten (“make yourself beloved there A.21. as you are everywhere”). “It was a ray of sunshine,” she recalled in Propositionin 4.11.5 (Convergence except on a set of measure 0). n a note published in Message d’un If fk is a series of Riemann-integrable functions on R such that math ematicien: Henri Lebesgue, X∞ Z pour le centenaire de sa naissance, n |fk(x)||d x| = A<∞, 4.11.12 Rn Paris, A. Blanchard, 1974. k=1 X∞ then fk(x) converges except for x in a set X of measure 0. k=1
Recall that “except on a set of measure 0” is also written “almost ev- erywhere” (or a.e.). So above we could simply say that the sum converges almost everywhere. 4.11 Lebesgue integrals 499
We can now de ne “equal in the sense of Lebesgue,” denoted =. L
De nitionin 4.11.6 (Lebesgue equality). Let f and g be two se- This notion of “Lebesgue equal- k k ity” is fairly subtle, as sets of mea- quences of R-integrable functions such that Z Z sure 0 can be quite complicated. X∞ X∞ n n For instance, if you only know a |fk(x)||d x| < ∞ and |gk(x)||d x| < ∞. Rn Rn function almost everywhere, then k=1 k=1 you can never evaluate it at any We will say that point: you never know whether ∞ ∞ ∞ ∞ this is a point at which you know X X X X fk = gk if fk(x)= gk(x) a.e. 4.11.13 the function. The moral: func- L tions that you know except on a k=1 k=1 k=1 k=1 set of measure 0 should only ap- pear under integral signs. Theorem 4.11.7 will make it possible to de ne the Lebesgue integral. Lebesgue integration is supe- rior to Riemann integration. It Theoremin 4.11.7. Let fk and gk be two sequences of R-integrable func- does not require functions to be tions such that bounded with bounded support, it X∞ Z X∞ Z ignores “local nonsense” on sets n n |fk(x)||d x| < ∞, |gk(x)||d x| < ∞, 4.11.14 of measure 0, and it is better be- Rn Rn k=1 k=1 haved with respect to limits. But if you want to compute in- and tegrals, the Riemann integral is X∞ X∞ still essential. Lebesgue integrals fk = gk. 4.11.15 L are more or less uncomputable un- k=1 k=1 less you know a function as a limit Then of Riemann-integrable functions in X∞ Z X∞ Z an appropriate sense – in the sense n n fk(x)|d x| = gk(x)|d x|. 4.11.16 of proposition 4.11.5, for instance. Rn Rn k=1 k=1
Thus the integral of a function f that is the sum of a series of R-integrable functions as in 4.11.12 depends only on f and not on the series. So we can now de ne the Lebesgue integral. Note that the series on the right of equation 4.11.18 is convergent, De nitionin 4.11.8 (Lebesgue integral). Let fk be a sequence of R- since (by part 4 of proposition integrable functions such that 4.1.13) it is absolutely convergent: ∞ Z Z X | || n | ∞ n fk(x) d x < . 4.11.17 fk(x)|d x| Rn k=1 Rn Z ∞ n X |fk(x)||d x| < ∞. Rn Then the Lebesgue integral of f = fk is k=1 Z X∞ Z n n f(x)|d x| = fk(x)|d x|. 4.11.18 Rn Rn k=1