The Fundamental Theorem of Calculus for Lebesgue Integral

Transcription

1 Divulgciones Mtemátics Vol. 8 No. 1 (2000), pp The Fundmentl Theorem of Clculus for Lebesgue Integrl El Teorem Fundmentl del Cálculo pr l Integrl de Lebesgue Diómedes Bárcens (brcens@ciens.ul.ve) Deprtmento de Mtemátics. Fcultd de Ciencis. Universidd de los ndes. Mérid. Venezuel. bstrct In this pper we prove the Theorem nnounced in the title without using Vitli s Covering Lemm nd hve s consequence of this pproch the equivlence of this theorem with tht which sttes tht bsolutely continuous functions with zero derivtive lmost everywhere re constnt. We lso prove tht the decomposition of bounded vrition function is unique up to constnt. Key words nd phrses: Rdon-Nikodym Theorem, Fundmentl Theorem of Clculus, Vitli s covering Lemm. Resumen En este rtículo se demuestr el Teorem Fundmentl del Cálculo pr l integrl de Lebesgue sin usr el Lem del cubrimiento de Vitli, obteniéndose como consecuenci que dicho teorem es equivlente l que firm que tod función bsolutmente continu con derivd igul cero en csi todo punto es constnte. Tmbién se prueb que l descomposición de un función de vrición cotd es únic menos de un constnte. Plbrs y frses clve: Teorem de Rdon-Nikodym, Teorem Fundmentl del Cálculo, Lem del cubrimiento de Vitli. Received: 1999/08/18. Revised: 2000/02/24. ccepted: 2000/03/01. MSC (1991): 2624, Supported by C.D.C.H.T-U.L. under project C

2 76 Diómedes Bárcens 1 Introduction The Fundmentl Theorem of Clculus for Lebesgue Integrl sttes tht: function f : [, b] R is bsolutely continuous if nd only if it is differentible lmost everywhere, its derivtive f L 1 [, b] nd, for ech t [, b], f(t) = f() + t f (s)ds. This theorem is extremely importnt in Lebesgue integrtion Theory nd severl wys of proving it re found in clssicl Rel nlysis. One of the better known proofs relies on the non trivil Vitli Covering Lemm, perhps influenced by Sks monogrphy [9]; we recommend Gordon s book [5] s recent reference. There re other pproches to the subject voiding Vitli s Covering Lemm, such s using the Riesz Lemm: Riez-Ngy [7] is the clssicl reference. nother pproch cn be seen in Rudin ([8], chpter VIII) which trets the subject by differentiting mesures nd of course mkes use of Lebesgue Decomposition nd the Rdon-Nikodym Theorem. The usul form of proving this fundmentl result runs more or less s follows: First of ll, Lebesgue Differentition Theorem is estblished: Every bounded vrition function f : [, b] R is differentible lmost everywhere with derivtive belonging to L 1 [, b]. If the function f is nondecresing, then b f (s)ds f(b) f(). In the clssicl proof of the bove theorem Vitli s Covering Lemm (Riesz Lemm either) is used, s well s in tht of the following Lemm, previous to the proof of the theorem we re interested in. If f : [, b] R is bsolutely continuous with f = 0 lmost everywhere then f is constnt. n elementry nd elegnt proof of this Lemm using tgged prtitions hs recently been chieved by Gordon [6]. In this pper we present nother pproch to the subject; indeed, we strt with Lebesgue Decomposition nd Rdon Nikodym Theorem nd soon fter, we derive directly the Fundmentl Theorem of Clculus nd get reltively simple proofs of well known results.

3 The Fundmentl Theorem of Clculus for Lebesgue Integrl 77 We strt outlining the proof of the Rdon Nikodym Theorem given by Brdley [4] in slightly different wy; indeed, insted of giving the proof of Rdon Nikodym Theorem s in [4], we mke direct (shorter) proof of the Lebesgue Decomposition Theorem which hs s corollry the Rdon Nikodym Theorem. Reders fmilirized with probbility theory will soon recognize the presence of mrtingle theory in this pproch. The needed preliminries for this pper re ll studied in regulr grdute course in Rel nlysis, especilly we need the Monotone nd Dominted Convergence theorems, the Cuchy-Schwrtz inequlity (n elementry proof is provided by postol [1]), the fct tht if (, Σ, µ) is mesure spce nd g L 1 (µ), then ν : Σ R defined for ν(e) = g dµ is rel mesure nd E f L 1 (ν) if nd only if fg L 1 (µ) nd E f dν = E fg dµ, E Σ. We lso need the definitions of mutully orthogonl mesures (µ λ) nd mesure λ bsolutely continuous with respect to µ (λ µ). While these preliminry fcts, previous to Lebesgue Decomposition Theorem nd Rdon Nikodym Theorem, re nicely treted in Rudin [8], good ccount of distribution functions (bounded vrition functions) cn be found in Burrill [3]. Prticulrly importnt re the following results: Every bounded vrition function f : [, b] R determines unique Lebesgue-Stieljes mesure µ. The function f is bsolutely continuous if nd only if its corresponding Lebesgue-Stieljes mesure µ is bsolutely continuous with respect to Lebesgue mesure. It is lso importnt for our purposes the following fct: If f : [, b] R is bounded vrition function with ssocited Lebesgue- Stieljes mesure µ, then the following sttements re equivlent: ) f is differentible t x nd f (x) =. b) For ech ɛ > 0 there is δ > 0 such tht µ(i) m(i) < ɛ, whenever I is n open intervl with Lebesgue mesure m(i) < δ nd x I, Both references Rudin [8] nd Burrill [3] re worth to look for the proof of this equivlence.

4 78 Diómedes Bárcens 2 Lebesgue Decomposition nd Rdon-Nikodym Theorem The Rdon Nikodym Theorem plys key role in our proof of the Fundmentl Theorem of Clculus, prticulrly the proof given by Brdley [4], so we will outline this proof but deriving it from Lebesgue Decomposition Theorem (Theorem 1 below). Theorem 1. (Lebesgue Decomposition Theorem). Let (, Σ, µ) be finite, positive mesure spce nd λ : R bounded vrition mesure. Then there is unique pir of mesures λ nd λ s so tht λ = λ + λ s with λ µ nd λ s µ. Proof. We first prove the uniqueness: if λ 1 + λ 1 s = λ 2 + λ 2 s with then λ i µ; λ i s µ qqudi = 1, 2, λ 1 λ 2 = λ 2 s λ 1 s with both sides of this eqution simultneously bsolutely continuous nd singulr with respect to µ. This quickly yields λ 1 = λ 2 nd λ 1 s = λ 2 s. Existence: This portion of the proof is modelled on Brdley s ide [4]. Let us denote by λ the vrition of λ nd put σ = µ + λ. Then µ nd λ re bsolutely continuous with respect to the finite positive mesure σ. Now for ny mesurble finite prtition P = { 1, 2,..., n } of we define where c i = h P = c i χ i, i=1 { 0 if σ(i ) = 0, λ( i) σ( i ) otherwise. (Our proof omits some detils which cn be found in [4]). Notice tht the following three sttements hold for ech finite mesurble prtition P :

5 The Fundmentl Theorem of Clculus for Lebesgue Integrl 79 ) 0 h P (x) 1. b) λ() = h P dσ. c) If P nd Π re finite mesurble prtitions of with Π refinement of P, then h 2 Π dσ = h 2 P dσ + (h Π h P ) 2 dσ h 2 P dσ. If we put k = sup h 2 P d σ, where the supremum is tken over ll finite mesurble prtitions of, then, since for ny finite mesurble prtition P of we hve tht h 2 P dσ = = i=1 i=1 λ( i ) 2 σ( i ) 2 dσ λ( i ) 2 σ( i ) λ( i ) i=1 λ () <, we conclude tht 0 k <. For ech n = 1, 2,... tke s P n finite mesurble prtition of such tht k 1 4 n h 2 P n dσ nd let Π n be the lest common refinement of P 1, P 2,..., P n. By (c), k 1 4 n h 2 P n dσ h 2 Π n dσ k.

6 80 Diómedes Bárcens So for ech n > 1, we hve (h Πn+1 h Πn ) 2 dσ = h 2 Π n+1 dσ h 2 Π n dσ 1 4 n nd by the Cuchy-Schwrtz inequlity we get h Πn+1 h Πn dσ 1 2 n (σ()) 1 2 < nd so which implies tht h Πn+1 h Πn dσ (σ()) 1 2 < i=1 h Πn+1 = h Π1 + h Πi+1 h Πi converges σ-lmost everywhere. Put { lim n h Πn (x) if the limits exists, h(x) = 0 otherwise. If Σ nd n N tke R n s the smllest common refinement of Π n nd {, \} to get h Rn h Πn 2 dσ 1 4 n. By (c) nd the Cuchy-Schwrtz inequlity, for n 1, (h Rn h Πn ) 2 dσ h Rn h Πn 2 dσ < 1 2 n, which implies tht Since by (b), λ() = i=1 lim (h Rn h Πn ) dσ = 0. n h Πn dσ + (h Rn h Πn ) dσ,

7 The Fundmentl Theorem of Clculus for Lebesgue Integrl 81 we conclude tht λ() = lim h Πn dσ n nd pplying the Dominted Convergence Theorem, we get λ() = h dσ Σ. Summrizing we hve gotten σ-integrble function h λ, depending on λ, such tht λ() = h λ dσ, Σ. Similrly we cn find σ-integrble function h µ (depending on µ) such tht µ() = h µ dσ Σ. Put 1 = {x : h µ (x) 0}, 2 = {x : h µ (x) > 0}. Plinly µ( 1 ) = 0 nd defining, for Σ, λ s () = λ( 1 ) nd λ () = λ( 2 ), we see tht λ s µ nd λ + λ s = λ. It remins to prove tht λ µ. Put h λ (x) if x 2 h µ (x) h(x) = 0 if x 1. Then for ny mesurble set 2 we hve λ () = λ() = h λ dσ = hh µ dσ = which implies λ µ. h dµ,

8 82 Diómedes Bárcens From Lebesgue Decomposition Theorem (nd its proof) we get Corollry 2. (Rdon Nikodym Theorem) If in the former theorem λ µ, then there is unique h L 1 (µ) such tht λ() = h dµ, Σ. 3 The Fundmentl Theorem of Clculus Now we re in position to prove the theorem we re interested in. Let f : [, b] R be n bsolutely continuous function nd µ its corresponding Lebesgue-Stieljes mesure. Then µ is bsolutely continuous with respect to the Lebesgue mesure m. By the Rdon-Nikodym Theorem there exists h L 1 (m) such tht µ() = h dm, Σ. Proof. Let us define the sequence of prtitions P n s Now define nd notice tht P n = {[x i, x i+1 )} 2n i=1, x i = + i (b ). 2n 2 n µ( i ) m( h n (x) = i=1 i ) χ i(x) if x b, 0 if x = b. lim h n(x) = h(x) n m-lmost everywhere. Hence f is differentible lmost everywhere nd f (x) = h(x) m-.e. Furthermore for ech t [, b], f(t) f() = µ([, t]) = t h(s)ds = t f (s)ds. So the Fundmentl Theorem of Clculus for Lebesgue integrl hs been proved.

9 The Fundmentl Theorem of Clculus for Lebesgue Integrl 83 The proof of the following corollry cn t be esier. Corollry 3. If f is bsolutely continuous nd f = 0 m-lmost everywhere, then f is constnt. Since clssicl proofs of the Fundmentl Clculus Theorem use the bove corollry we conclude the following result: Theorem 4. The Following sttements re equivlent ) Every bsolutely continuous function for which f = 0 m-.e. is constnt function. b) If f : [, b] R is bsolutely continuous then f is differentible lmost everywhere with f(t) f() = t f (s) ds, t [, b]. nother old gem from Lebesgue Integrtion Theory is the following one. Corollry 5. If f : [, b] R is Lebesgue integrble (regrding Lebesgue mesure) nd g(t) = t f(s) ds (t [, b]), then g is differentible lmost everywhere nd g (s) = f(s) for lmost every t [, b]. Proof. Using stndrd techniques we estblish the bsolute continuity of g. So g is differentible lmost everywhere nd g(t) = t g (s) ds for ech t [, b]. From this nd the definition of g, we conclude tht g (s) = f(s) lmost everywhere. Lebesgue Differentition Theorem hs been proved by ustin [2] without using Vitli s Covering Lemm. Becuse differentible functions hve mesurble derivtives, this llows us to get the following theorem without using Vitli s Lemm. Recll tht function f : [, b] R is singulr if nd only if it hs zero derivtive lmost everywhere ([8]). Theorem 6. If f : [, b] R is bounded vrition function, then it is the sum of n bsolutely continuous function plus singulr function. This Decomposition is unique up to constnt. Proof. Uniqueness: Suppose f = f 1 + f 2 = g 1 + g 2

10 84 Diómedes Bárcens with f 1, g 1 singulr nd f 2, g 2 bsolutely continuous functions. Then f 1 g 1 = g 2 f 2 is both bsolutely continuous nd singulr. These fcts imply tht (f 2 g 2 ) = 0 lmost everywhere nd so f 2 g 2 is constnt. Existence: Since f is of bounded vrition, then f exists lmost everywhere. If f is non-decresing, it is not hrd to see tht for ech t [, b] f L 1 [, b] nd 0 t 0 f (s)ds f(t) f() nd, since ny bounded vrition function is the difference of two non-decresing functions, we relize tht f L 1 [, b]. If we define for t [, b] h(t) = t f (s)ds + f(), then h is bsolutely continuous nd g(t) = f(t) h(t) is singulr with f = g+h. This ends the proof. Remrk. Regrding the bove Theorem, this is the only proof tht we know bout uniqueness. References [1] postol, T., Clculus, vol.2, Blisdell, Wlthm, [2] ustin. D., Geometric proof of the Lebesgue Differentition Theorem, Proc. mer. Mth. Soc. 16 (1965), [3] Burrill, C. W., Mesure, Integrtion nd Probbility, McGrw Hill, New York, [4] Brdley, R. C., n Elementry Tretment of the Rdon-Nikodym Derivtive, mer. Mth. Monthly 96(5) (1989), [5] Gordon, R., The integrls of Lebesgue, Denjoy, Perron, nd Henstock, Grdute Studies in Mthemtics, vol. 4, mer. Mth. Soc., Providence, RI, 1994.

11 The Fundmentl Theorem of Clculus for Lebesgue Integrl 85 [6] Gordon, R.., The use of tgged prtition in Elementry Rel nlysis, mer. Mth. Monthly, 105(2) (1998), [7] Riez, F., Ngy, B. Sz., Leçons d nlyse Fonctionnelle, kdémii Kidó, Budpest, [8] Rudin, W., Rel nd Complex nlysis, 2nd edition, McGrw Hill, New York, [9] Sks S., The Theory of Integrl, Dover, New York, 1964.