Version of 24.2.4 Chpter 22 The Fundmentl Theorem of Clculus In this chpter I ddress one of the most importnt properties of the Lebesgue integrl. Given n integrble function f : [,b] R, we cn form its indefinite integrl F(x) = x f(t)dt for x [,b]. Two questions immeditely present themselves. (i) Cn we expect to hve the derivtive F of F equl to f? (ii) Cn we identify which functions F will pper s indefinite integrls? Resonbly stisfctory nswers my be found for both of these questions: F = f lmost everywhere (222E) nd indefinite integrls re the bsolutely continuous functions (225E). In the course of deling with them, we need to develop vriety of techniques which led to mny striking results both in the theory of Lebesgue mesure nd in other, pprently unrelted, topics in rel nlysis. The first step is Vitli s theorem ( 22), remrkble rgument it is more method thn theorem which uses the geometric nture of the rel line to extrct disjoint subfmilies from collections of intervls. It is the foundtion stone not only of the results in 222 but of ll geometric mesure theory, tht is, mesure theory on spces with geometric structure. I use it here to show tht monotonic functions re differentible lmost everywhere (222A). Following this, Ftou s Lemm nd Lebesgue s Dominted Convergence Theorem re enough to show tht the derivtive of n indefinite integrl is lmost everywhere equl to the integrnd. We find tht some innocent-looking mnipultions of this fct tke us surprisingly fr; I present these in 223. I begin the second hlf of the chpter with discussion of functions of bounded vrition, tht is, expressible s the difference of bounded monotonic functions ( 224). This is one of the lest mesure-theoretic sections in the volume; only in 224I nd 224J re mesure nd integrtion even mentioned. But this mteril isneededforchpter28swellsforthenextsection, ndislsooneofthebsictopicsoftwentieth-century rel nlysis. 225 dels with the chrcteriztion of indefinite integrls s the bsolutely continuous functions. In fct this is now quite esy; it helps to cll on Vitli s theorem gin, but everything else is strightforwrd ppliction of methods previously used. The second hlf of the section introduces some new ides in n ttempt to give deeper intuition into the essentil nture of bsolutely continuous functions. 226 returns to functions of bounded vrition nd their decomposition into sltus nd bsolutely continuous nd singulr prts, the first two being reltively mngeble nd the lst looking something like the Cntor function. 22 Vitli s theorem in R Version of 2.6.03 I give the first theorem of this chpter section to itself. It occupies position between mesure theory nd geometry (it is, indeed, one of the fundmentl results of geometric mesure theory ), nd its proof involves both the mesure nd the geometry of the rel line. 22A Vitli s theorem Let A be bounded subset of R nd I fmily of non-singleton closed intervls in R such tht every point of A belongs to rbitrrily short members of I. Then there is countble set I 0 I such tht (i) I 0 is disjoint, tht is, I I = for ll distinct I, I I 0 (ii) µ(a\ I 0 ) = 0, where µ is Lebesgue mesure on R. Extrct from Mesure Theory, results-only version, by D.H.Fremlin, University of Essex, Colchester. This mteril is copyright. It is issued under the terms of the Design Science License s published in http://dsl.org/copyleft/dsl.txt. This is development version nd the source files re not permnently rchived, but current versions re normlly ccessible through http://www.essex.c.uk/mths/people/fremdh/mt.htm. For further informtion contct fremdh@essex.c.uk. c 995 D. H. Fremlin
2 The Fundmentl Theorem of Clculus 222 Differentiting n indefinite integrl Version of 20..03/8.0.04 I come now to the first of the two questions mentioned in the introduction to this chpter: if f is n d x integrble function on [, b], wht is dx f? It turns out tht this derivtive exists nd is equl to f lmost everywhere (222E). The rgument is bsed on striking property of monotonic functions: they re differentible lmost everywhere (222A), nd we cn bound the integrls of their derivtives (222C). 222A Theorem Let I R be n intervl nd f : I R monotonic function. Then f is differentible lmost everywhere in I. 222B Remrks If (X,Σ,µ) is mesure spce, K is countble set, nd E k k K is fmily in Σ, with equlity if E k k K is disjoint. µ( k K E k) k K µe k, 222C Lemm Suppose tht b in R, nd tht F : [,b] R is non-decresing function. Then b F exists nd is t most F(b) F(). Remrk I write x f to men [,x[ f. 222D Lemm Suppose tht < b in R, nd tht f, g re rel-vlued functions, both integrble over [,b], such tht x f = x g for every x [,b]. Then f = g lmost everywhere in [,b]. 222E Theorem Suppose tht b in R nd tht f is rel-vlued function which is integrble over [,b]. Then F(x) = x f exists in R for every x [,b], nd the derivtive F (x) exists nd is equl to f(x) for lmost every x [,b]. 222F Corollry Suppose tht f is ny rel-vlued function which is integrble over R, nd set F(x) = x f for every x R. Then F (x) exists nd is equl to f(x) for lmost every x R. 222G Corollry Suppose tht E R is mesurble set nd tht f is rel-vlued function which is integrble over E. Set F(x) = E ],x[ f for x R. Then F (x) = f(x) for lmost every x E, nd F (x) = 0 for lmost every x R\E. 222H Proposition Suppose tht b in R nd tht f is rel-vlued function which is integrble over [,b]. Set F(x) = x f for x [,b]. Then F (x) exists nd is equl to f(x) t ny point x dom(f) ],b[ t which f is continuous. 222I Complex-vlued functions () If b in R nd f is complex-vlued function which is integrble over [,b], then F(x) = x f is defined in C for every x [,b], nd its derivtive F (x) exists nd is equl to f(x) for lmost every x [,b]; moreover, F (x) = f(x) whenever x dom(f) ],b[ nd f is continuous t x. (b) If f is complex-vlued function which is integrble over R, nd F(x) = x f for ech x R, then F exists nd is equl to f lmost everywhere in R. (c) If E R is mesurble set nd f is complex-vlued function which is integrble over E, nd F(x) = E ],x[ f for ech x R, then F (x) = f(x) for lmost every x E nd F (x) = 0 for lmost every x R\E. c 2004 D. H. Fremlin Mesure Theory (bridged version)
223A Lebesgue s density theorems 3 *222J The Denjoy-Young-Sks theorem: Definition Let f be ny rel function, nd A R its domin. Write à + = {x : x A, ]x,x+δ] A for every δ > 0}, Set for x Ã+, nd à = {x : x A, [x δ,x[ A for every δ > 0}. D + (x) = limsup y A,y x D + (x) = liminf y A,y x D (x) = limsup y A,y x = inf δ>0 sup y A,x<y x+δ, = sup δ>0 inf y A,x<y x+δ = inf δ>0 sup y A,x δ y<x, D (x) = liminf y A,y x = sup δ>0 inf y A,x δ y<x for x Ã, ll defined in [, ]. (These re the four Dini derivtes of f.) Note tht we surely hve (D + f)(x) (D + f)(x) for every x Ã+, while (D f)(x) (D f)(x) for every x Ã. The ordinry derivtive f (x) is defined nd equl to c R iff (α) x belongs to some open intervl included in A (β) (D + f)(x) = (D + f)(x) = (D f)(x) = (D + f)(x) = c. *222K Lemm Let A be ny subset of R, nd define Ã+ nd à s in 222J. Then A\Ã+ nd A\à re countble, therefore negligible. *222L Theorem Let f be ny rel function, nd A its domin. Then for lmost every x A either ll four Dini derivtes of f t x re defined, finite nd equl or (D + f)(x) = (D f)(x) is finite, (D + f)(x) = nd (D + f)(x) = or (D + f)(x) = (D f)(x) is finite, (D + f)(x) = nd (D f)(x) = or (D + f)(x) = (D f)(x) = nd (D + f)(x) = (D f)(x) =. 223 Lebesgue s density theorems Version of 9.9.04 I now turn to group of results which my be thought of s corollries of Theorem 222E, but which lso hve vigorous life of their own, including the possibility of significnt generliztions which will be treted in Chpter 26. The ide is tht ny mesurble function f on R is lmost everywhere continuous in vriety of very wek senses; for lmost every x, the vlue f(x) is determined by the behviour of f ner x, in the sense tht f(y) f(x) for most y ner x. I should perhps sy tht while I recommend this work s preprtion for Chpter 26, nd I lso rely on it in Chpter 28, I shll not refer to it gin in the present chpter, so tht reders in hurry to chrcterize indefinite integrls my proceed directly to 224. 223A Lebesgue s Density Theorem: integrl form Let I be n intervl in R, nd let f be rel-vlued function which is integrble over I. Then for lmost every x I. f(x) = lim h 0 h x x f = lim h 0 h f = lim h 0 f c 995 D. H. Fremlin D.H.Fremlin
4 The Fundmentl Theorem of Clculus 223B 223B Corollry Let E R be mesurble set. Then lim h 0 µ(e [,x+h]) = for lmost every x E, lim h 0 µ(e [,x+h]) = 0 for lmost every x R\E. 223C Corollry Let f be mesurble rel-vlued function defined lmost everywhere in R. Then for lmost every x R, lim h 0 µ{y : y domf, y x h, ǫ} =, for every ǫ > 0. lim h 0 µ{y : y domf, y x h, ǫ} = 0 223D Theorem Let I be n intervl in R, nd let f be rel-vlued function which is integrble over I. Then lim h 0 dy = 0 for lmost every x I. Remrk The set {x : x domf, lim h 0 is sometimes clled the Lebesgue set of f. 223E Complex-vlued functions dy = 0} () Let I be n intervl in R, nd let f be complex-vlued function which is integrble over I. Then for lmost every x I. f(x) = lim h 0 h x x f = lim h 0 h f = lim h 0 f (b) Let f be mesurble complex-vlued function defined lmost everywhere in R. Then for lmost every x R, for every ǫ > 0. lim h 0 µ{y : y domf, y x h, ǫ} = 0 (c) Let I be n intervl in R, nd let f be complex-vlued function which is integrble over I. Then for lmost every x I. lim h 0 dy = 0 Version of 29.9.04 224 Functions of bounded vrition c 997 D. H. Fremlin Mesure Theory (bridged version)
224F Functions of bounded vrition 5 I turn now to the second of the two problems to which this chpter is devoted: the identifiction of those rel functions which re indefinite integrls. I tke the opportunity to offer brief introduction to the theory of functions of bounded vrition, which re interesting in themselves nd will be importnt in Chpter 28. I give the bsic chrcteriztion of these functions s differences of monotonic functions (224D), with representtive smple of their elementry properties. 224A Definition Let f be rel-vlued function nd D subset of R. I define Vr D (f), the (totl) vrition of f on D, s follows. If D domf =, Vr D (f) = 0. Otherwise, Vr D (f) is sup{ n i= f( i) f( i ) : 0,,..., n D domf, 0... n }, llowing Vr D (f) =. If Vr D (f) is finite, we sy tht f is of bounded vrition on D. I my write Vrf for Vr domf (f), nd sy tht f is simply of bounded vrition if this is finite. 224B Remrks for ll D, f. Vr D (f) = Vr D domf (f) = Vr(f D) 224C Proposition () If f, g re two rel-vlued functions nd D R, then Vr D (f +g) Vr D (f)+vr D (g). (b) If f is rel-vlued function, D R nd c R then Vr D (cf) = c Vr D (f). (c) If f is rel-vlued function, D R nd x R then Vr D (f) Vr D ],x] (f)+vr D [x, [ (f), with equlity if x D domf. (d) If f is rel-vlued function nd D D R then Vr D (f) Vr D (f). (e) If f is rel-vlued function nd D R, then f(x) f(y) Vr D (f) for ll x, y D domf; so if f is of bounded vrition on D then f is bounded on D domf nd (if D domf ) sup y D domf f(y) f(x) +Vr D (f) for every x D domf. (f) If f is monotonic rel-vlued function nd D R meets domf, then Vr D (f) = sup x D domf f(x) inf x D domf f(x). 224D Theorem For ny rel-vlued function f nd ny set D R, the following re equiveridicl: (i) there re two bounded non-decresing functions f, f 2 : R R such tht f = f f 2 on D domf; (ii) f is of bounded vrition on D; (iii) there re bounded non-decresing functions f, f 2 : R R such tht f = f f 2 on D domf nd Vr D (f) = Vrf +Vrf 2. 224E Corollry Let f be rel-vlued function nd D ny subset of R. If f is of bounded vrition on D, then for every R, nd lim x Vr D ],x] (f) = lim x Vr D [x,[ (f) = 0 lim Vr D ],] (f) = lim Vr D [, [ (f) = 0. 224F Corollry Let f be rel-vlued function of bounded vrition on [,b], where < b. If domf meets every intervl ],+δ] with δ > 0, then lim t domf,t f(t) D.H.Fremlin
6 The Fundmentl Theorem of Clculus 224F is defined in R. If domf meets [b δ,b[ for every δ > 0, then is defined in R. lim t domf,t b f(t) 224G Corollry Let f, g be rel functions nd D subset of R. If f nd g re of bounded vrition on D, so is f g. 224H Proposition Let f : D R be function of bounded vrition, where D R. Then f is continuous t ll except countbly mny points of D. 224I Theorem Let I R be n intervl, nd f : I R function of bounded vrition. Then f is differentible lmost everywhere in I, nd f is integrble over I, with I f Vr I (f). 224J Proposition Let f, g be rel-vlued functions defined on subsets of R, nd suppose tht g is integrble over n intervl [,b], where < b, nd f is of bounded vrition on ],b[ nd defined lmost everywhere in ],b[. Then f g is integrble over [,b], nd b f g ( lim f(x) +Vr ],b[(f) ) sup x domf,x b c [,b] c g. 224K Complex-vlued functions () Let D be subset of R nd f complex-vlued function. The vrition of f on D, Vr D (f), is zero if D domf =, nd otherwise is sup{ n j= f( j) f( j ) : 0... n in D domf}, llowing. If Vr D (f) is finite, we sy tht f is of bounded vrition on D. (b) A complex-vlued function of bounded vrition must be bounded, nd for every x R, with equlity if x D domf, Vr D (f +g) Vr D (f)+vr D (g), Vr D (cf) = c Vr D (f), Vr D (f) Vr D ],x] (f)+vr D [x, [ (f) Vr D (f) Vr D (f) whenever D D. (c) A complex-vlued function is of bounded vrition iff its rel nd imginry prts re both of bounded vrition. So complex-vlued function f is of bounded vrition on D iff there re bounded non-decresing functions f,...,f 4 : R R such tht f = f f 2 +if 3 if 4 on D. (d) Let f be complex-vlued function nd D ny subset of R. If f is of bounded vrition on D, then for every R, nd lim x Vr D ],x] (f) = lim x Vr D [x,[ (f) = 0 lim Vr D ],] (f) = lim Vr D [, [ (f) = 0. (e) Let f be complex-vlued function of bounded vrition on [,b], where < b. If domf meets every intervl ],+δ] with δ > 0, then lim t domf,t f(t) is defined in C. If domf meets [b δ,b[ for every δ > 0, then lim t domf,t b f(t) is defined in C. Mesure Theory (bridged version)
225F Absolutely continuous functions 7 (f) Let f, g be complex functions nd D subset of R. If f nd g re of bounded vrition on D, so is f g. (g) Let I R be n intervl, nd f : I C function of bounded vrition. Then f is differentible lmost everywhere in I, nd I f Vr I (f). (h) Let f nd g be complex-vlued functions defined on subsets of R, nd suppose tht g is integrble over n intervl [,b], where < b, nd f is of bounded vrition on ],b[ nd defined lmost everywhere in ],b[. Then f g is integrble over [,b], nd b f g ( lim f(x) +Vr ],b[(f) ) sup x domf,x b c [,b] c g. Version of 6.8.5 225 Absolutely continuous functions We re now redy for full chrcteriztion of the functions tht cn pper s indefinite integrls (225E, 225Xf). The essentil ide is tht of bsolute continuity (225B). In the second hlf of the section (225G-225N) I describe some of the reltionships between this concept nd those we hve lredy seen. 225A Absolute continuity of the indefinite integrl: Theorem Let (X, Σ, µ) be ny mesure spce nd f n integrble rel-vlued function defined on conegligible subset of X. Then for ny ǫ > 0 there re mesurble set E of finite mesure nd rel number δ > 0 such tht f ǫ whenever F Σ nd F µ(f E) δ. 225B Absolutely continuous functions on R: Definition If [, b] is non-empty closed intervl in R nd f : [,b] R is function, we sy tht f is bsolutely continuous if for every ǫ > 0 there is δ > 0 such tht n i= f(b i) f( i ) ǫ whenever b 2 b 2... n b n b nd n i= b i i δ. 225C Proposition Let [,b] be non-empty closed intervl in R. () If f : [,b] R is bsolutely continuous, it is uniformly continuous. (b) If f : [,b] R is bsolutely continuous it is of bounded vrition on [,b], so is differentible lmost everywhere in [, b], nd its derivtive is integrble over [, b]. (c) If f, g : [,b] R re bsolutely continuous, so re f +g nd cf, for every c R. (d) If f, g : [,b] R re bsolutely continuous so is f g. (e) If g : [,b] [c,d] nd f : [c,d] R re bsolutely continuous, nd g is non-decresing, then the composition fg : [,b] R is bsolutely continuous. 225D Lemm Let [,b] be non-empty closed intervl in R nd f : [,b] R n bsolutely continuous function which hs zero derivtive lmost everywhere in [,b]. Then f is constnt on [,b]. 225E Theorem Let [,b] be non-empty closed intervl in R nd F : [,b] R function. Then the following re equiveridicl: (i) there is n integrble rel-vlued function f such tht F(x) = F()+ x f for every x [,b]; (ii) x F exists nd is equl to F(x) F() for every x [,b]; (iii) F is bsolutely continuous. 225F Integrtion by prts: Theorem Let f be rel-vlued function which is integrble over n intervl [,b] R, nd g : [,b] R n bsolutely continuous function. Suppose tht F is n indefinite integrl of f, so tht F(x) F() = x f for x [,b]. Then b f g = F(b)g(b) F()g() b F g. c 996 D. H. Fremlin D.H.Fremlin
8 The Fundmentl Theorem of Clculus 225G 225G Proposition Let [,b] be non-empty closed intervl in R nd f : [,b] R n bsolutely continuous function. () f[a] is negligible for every negligible set A R. (b) f[e] is mesurble for every mesurble set E R. 225H Semi-continuous functions If D R r, function g : D [, ] is lower semi-continuous if {x : g(x) > u} is n open subset of D for every u [, ]. Any lower semi-continuous function is Borel mesurble, therefore Lebesgue mesurble. 225I Proposition Suppose tht r nd tht f is rel-vlued function, defined on subset D of R r, which is integrble over D. Then for ny ǫ > 0 there is lower semi-continuous function g : R r [, ] such tht g(x) f(x) for every x D nd D g is defined nd not greter thn ǫ+ D f. 225J Theorem Let D be subset of R nd f : D R ny function. Then is reltively Borel mesurble in D, nd E = {x : x D, f is continuous t x} F = {x : x D, f is differentible t x} is Borel mesurble in R; moreover, f : F R is Borel mesurble. 225K Proposition Let [,b] be non-empty closed intervl in R, nd f : [,b] R function. Set F = {x : x ],b[, f (x) is defined}. Then f is bsolutely continuous iff (i) f is continuous (ii) f is integrble over F (iii) f[[,b]\f] is negligible. 225L Corollry Let [,b] be non-empty closed intervl in R. Let f : [,b] R be continuous function which is differentible on the open intervl ],b[. If its derivtive f is integrble over [,b], then f is bsolutely continuous, nd f(b) f() = b f. 225M Corollry Let [,b] be non-empty closed intervl in R, nd f : [,b] R continuous function. Then f is bsolutely continuous iff it is continuous nd of bounded vrition nd f[a] is negligible for every negligible A [,b]. 225N The Cntor function Let C [0,] be the Cntor set. Recll tht the Cntor function is non-decresing continuous function f : [0,] [0,] such tht f (x) is defined nd equl to zero for every x [0,] \ C, but f(0) = 0 < = f(). f is of bounded vrition nd not bsolutely continuous. C is negligible nd f[c] = [0,] is not. If x C, then for every n N there is n intervl of length 3 n, contining x, on which f increses by 2 n ; so f cnnot be differentible t x, nd the set F = domf of 225K is precisely [0,]\C, so tht f[[0,]\f] = [0,]. 225O Complex-vlued functions () Let (X, Σ, µ) be ny mesure spce nd f n integrble complexvlued function defined on conegligible subset of X. Then for ny ǫ > 0 there re mesurble set E of finite mesure nd rel number δ > 0 such tht f ǫ whenever F Σ nd µ(f E) δ. F (b) If [,b] is non-empty closed intervl in R nd f : [,b] C is function, we sy tht f is bsolutely continuous if for every ǫ > 0 there is δ > 0 such tht n i= f(b i) f( i ) ǫ whenever b 2 b 2... n b n b nd n i= b i i δ. Observe tht f is bsolutely continuous iff its rel nd imginry prts re both bsolutely continuous. (c) Let [,b] be non-empty closed intervl in R. (i) If f : [,b] C is bsolutely continuous it is of bounded vrition on [,b], so is differentible lmost everywhere in [, b], nd its derivtive is integrble over [, b]. (ii) If f, g : [,b] C re bsolutely continuous, so re f +g nd ζf, for ny ζ C, nd f g. (iii) If g : [, b] [c, d] is monotonic nd bsolutely continuous, nd f : [c, d] C is bsolutely continuous, then fg : [,b] C is bsolutely continuous. Mesure Theory (bridged version)
226Ad The Lebesgue decomposition of function of bounded vrition 9 (d) Let [,b] be non-empty closed intervl in R nd F : [,b] C function. Then the following re equiveridicl: (i) there is n integrble complex-vlued function f such tht F(x) = F()+ x f for every x [,b]; (ii) x F exists nd is equl to F(x) F() for every x [,b]; (iii) F is bsolutely continuous. (e) Let f be n integrble complex-vlued function on n intervl [,b] R, nd g : [,b] C n bsolutely continuous function. Set F(x) = x f for x [,b]. Then b f g = F(b)g(b) F()g() b F g. (f) Let f be continuous complex-vlued function on closed intervl [,b] R, nd suppose tht f is differentible t every point of the open intervl ],b[, with f integrble over [,b]. Then f is bsolutely continuous. 226 The Lebesgue decomposition of function of bounded vrition Version of 6..3 I end this chpter with some notes on method of nlysing generl function of bounded vrition which my help to give picture of wht such functions cn be, though (prt from 226A) it is hrdly needed in this volume. 226A Sums over rbitrry index sets () If I is ny set nd i i I ny fmily in [0, ], we set i I i = sup{ i K i : K is finite subset of I}, with the convention tht i i = 0. For generl i [, ], we cn set i I i = i I + i i I i if this is defined in [, ], where + = mx(,0) nd = mx(,0) for ech. If i I i is defined nd finite, we sy tht i i I is summble. (b) For ny set I, we hve the corresponding counting mesure µ on I. Every fmily i i I of rel numbers is mesurble rel-vlued function on I. A rel-vlued function f on I is simple if K = {i : f(i) 0} is finite. Now generl function f : I R is integrble iff i I f(i) <, nd in this cse fdµ = i I f(i), Thus we hve i I i = I iµ(di), nd the stndrd rules under which we llow s the vlue of n integrl mtch the interprettions in () bove. (c) I observe here tht this notion of summbility is bsolute ; fmily i i I is summble iff it is bsolutely summble. (d) If i i I is n (bsolutely) summble fmily of rel numbers, then for every ǫ > 0 there is finite K I such tht i I\K i ǫ. Consequently, for ny fmily i i I of rel numbers nd ny s R, the following re equiveridicl: (i) i I i = s; (ii) for every ǫ > 0 there is finite K I such tht s i J i ǫ whenever J is finite nd K J I. c 2000 D. H. Fremlin D.H.Fremlin
0 The Fundmentl Theorem of Clculus 226Ae (e) If i I i <, then J = {i : i 0} = n N {i : i 2 n } is countble. If J is finite, then i I i = i J i reduces to finite sum. Otherwise, we cn enumerte J s j n n N, nd we shll hve i I i = i J i = lim n n k=0 j k = n=0 j n. Conversely, if i i I is such tht there is countbly infinite J {i : i 0} enumerted s j n n N, nd if n=0 j n <, then i I i will be n=0 j n. (f) Let I nd J be sets nd ij i I,j J fmily in [0, ]. Then (i,j) I J ij = i I ( j J ij) = j J ( i I ij). 226B Sltus functions Suppose tht < b in R. () A (rel) sltus function on [,b] is function F : [,b] R expressible in the form F(x) = t [,x[ u t + t [,x] v t for x [,b], where u t t [,b[, v t t [,b] re rel-vlued fmilies such tht t [,b[ u t nd t [,b] v t re finite. (b) For ny function F : [,b] R we cn write F(x + ) = lim y x F(y) if x [,b[ nd the limit exists, F(x ) = lim y x F(y) if x ],b] nd the limit exists. Observe tht if F is sltus function, s defined in (b), with ssocited fmilies u t t [,b[ nd v t t [,b], then v = F(), v x = F(x) F(x ) for x ],b] nd u x = F(x + ) F(x) for x [,b[. F is continuous t x ],b[ iff u x = v x = 0, while F is continuous t iff u = 0 nd F is continuous t b iff v b = 0. In prticulr, {x : x [,b], F is not continuous t x} is countble. (c) If F is sltus function defined on [,b], with ssocited fmilies u t t [,b[ nd v t t [,b], then F is of bounded vrition on [,b], nd Vr [,b] (F) t [,b[ u t + t ],b] v t. (d) The inequlity in (c) is ctully n equlity. (e) Becuse sltus function is of bounded vrition, it is differentible lmost everywhere. In fct its derivtive is zero lmost everywhere. 226C The Lebesgue decomposition of function of bounded vrition Tke, b R with < b. () If F : [,b] R is non-decresing, set v = 0, v t = F(t) F(t ) for t ],b], u t = F(t + ) F(t) for t [,b[. Then ll the v t, u t re non-negtive, nd t [,b[ u t nd t [,b] v t re both finite. Let F p be the corresponding sltus function. F p nd F c = F F p re non-decresing. F c is continuous. Clerly this expression of F = F p +F c s the sum of sltus function nd continuous function is unique, except tht we cn freely dd constnt to one if we subtrct it from the other. (b) Set F c (x) = F()+ x F for ech x [,b]. F cs = F c F c is still non-decresing; F cs is continuous; F cs = 0.e. Agin, the expression of F c = F c +F cs s the sum of n bsolutely continuous function nd function with zero derivtive lmost everywhere is unique, except for the possibility of moving constnt from one to the other. Mesure Theory (bridged version)
226E The Lebesgue decomposition of function of bounded vrition (c) Putting these together: if F : [,b] R is ny non-decresing function, it is expressible s F p +F c + F cs, where F p is sltus function, F c is bsolutely continuous, nd F cs is continuous nd differentible, with zero derivtive, lmost everywhere; ll three components re non-decresing; nd the expression is unique if we sy tht F c () = F() nd F p () = F cs () = 0. The Cntor function f : [0,] [0,] is continuous nd f = 0.e., so f p = f c = 0 nd f = f cs. Setting g(x) = 2 (x+f(x)) for x [0,], we get g p(x) = 0, g c (x) = x 2 nd g cs(x) = 2 f(x). (d) Now suppose tht F : [,b] R is of bounded vrition. Then it is expressible s difference G H of non-decresing functions. So writing F p = G p H p, etc., we cn express F s sum F p + F cs + F c, where F p is sltus function, F c is bsolutely continuous, F cs is continuous, F cs = 0.e., F c () = F() nd F cs () = F p () = 0. Under these conditions the expression is unique. This is Lebesgue decomposition of the function F. I will cll F p the sltus prt of F. 226D Complex functions () If I is ny set nd j j I is fmily of complex numbers, then the following re equiveridicl: (i) j I j < ; (ii)thereisns Csuchthtforeveryǫ > 0thereisfiniteK I suchtht s j J j ǫ whenever J is finite nd K J I. In this cse s = j I Re( j)+i j I Im( j) = I jµ(dj), where µ is counting mesure on I, nd we write s = j I j. (b) If < b in R, complex sltus function on [,b] is function F : [,b] C expressible in the form F(x) = t [,x[ u t + t [,x] v t for x [,b], where u t t [,b[, v t t [,b] re complex-vlued fmilies such tht t [,b[ u t nd t [,b] v t re finite. In this cse F is continuous except t countbly mny points nd differentible, with zero derivtive, lmost everywhere in [, b], nd F is of bounded vrition, nd its vrition is u x = lim t x F(t) F(x) for every x [,b[, v x = lim t x F(x) F(t) for every x ],b]. Vr [,b] (F) = t [,b[ u t + t ],b] v t. (c) If F : [,b] C is function of bounded vrition, where < b in R, it is uniquely expressible s F = F p +F cs +F c, where F p is sltus function, F c is bsolutely continuous, F cs is continuous nd hs zero derivtive lmost everywhere, nd F c () = F(), F p () = F cs () = 0. 226E Proposition Let (X,Σ,µ) be mesure spce, I countble set, nd f i i I fmily of µ- integrble rel- or complex-vlued functions such tht i I fi dµ is finite. Then f(x) = i I f i(x) is defined lmost everywhere nd fdµ = i I fi dµ. D.H.Fremlin