DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda simple fuctios. This is a geeralizatio of the famous Lebesgue criterio of Riema itegrability. AMS 2000 Subject Classificatio: 46A40, 46A55. Key words: Cauchy-Daiell itegral, Riema itegrability. 1. PRELIMINARIES We cosider a liear vector space C of real bouded fuctios over a arbitrary set X, X φ, such that the costat real fuctios o X belog to C ad for ay f, g C the fuctio f g : X R defied as (f g(x = max{f(x, g(x} also belogs to C. Obviously, for ay f, g C the fuctio f g (respectively f defied as (f g(x = mi{f(x, g(x} (resp. f (x = f(x belogs to C ad we have f g + f g = f + g, f = ( f f. Defiitio 1. A real liear map I : C R is called a Daiell itegral (or a Cauchy-Daiell itegral if it is icreasig ad mootoe sequetially cotiuous, i.e., (a I (αf + βg = αi(f + βi(g α, β R, f, g C ; (b f, g C, f g I (f I(g; (c for ay decreasig sequece (f of C such that if {f (x N} = 0 for ay x X we have if I(f = 0. As is well ow, there exists a real positive measure µ o the σ-algebra B(C, geerated by C (the coarsest σ-algebra of sets o X for with respect to which ay fuctio f C is a measurable real fuctio o X, such that we have I(f = fdµ f C. REV. ROUMAINE MATH. PURES APPL., 54 (2009, 5 6, 417 422
418 Ileaa Bucur 2 There are several steps of extesio of the give fuctioal I such that, fially, ay characteristic fuctio 1 A with A B(C, belogs to the domai of the extesio of I. We recall briefly this procedure. First, we deote by C i (respectively, C s, the set of all fuctios ϕ : X (, ] (respectively, Ψ : X [,, for which there exists a icreasig sequece (f (respectively decreasig sequece (g i C, such that ϕ = sup f =: f (Ψ = if g =: g. Usig the property of a Daiell itegral oe ca show that the elemet I(ϕ (respectively, I(Ψ of (, ] (respectively, [, defied as I(ϕ = sup I(f (I(Ψ = if I(g does ot deped o the sequece (f (respectively, (g which icreases to ϕ (respectively, decreases to Ψ. We otice the followig facts: (1 C i ad C s are covex coes, i.e., for ay f, g i C i (respectively, C s ad for ay α, β R, α > 0, β > 0, we have αf + βg C i (respectively, C s. (2 for ay f, g i C i (respectively, C s the fuctios f g, f g belog to C i (respectively, C s. Moreover, for ay icreasig sequece of C i, its poitwise supremum belogs to C i while the poitwise ifimum of a decreasig sequece of C s belogs to C s. I fact, we have C s = C i, i.e., C s = { ϕ ϕ C i } or C i = { Ψ Ψ C s }. (3 I (αf + βg = αi(f + βi(g α, β R + ad f, g C i (respectively, f, g C s. (4 I(f I(g if f g ad f, g C i (respectively, f, g C s. (5 sup I(f = I ( sup f for ay icreasig sequece (f of C i ad if I(f = I ( if f for ay decreasig sequece (f of C s. (6 I ( f = I(f f C i or f C s. For ay fuctio h : X R, we deote by I (h (respectively, I (h the elemet of R defied as I (h = if {I(f f C i, f h} (respectively, I (h = sup {I(g g C s, g h}. The assertio I (h I (h does always hold. These ew extesios of I have the properties below: (7 I ad I are icreasig, i.e., I (f I (g, respectively, I (f I (g wheever f g; (8 I (h + g I (h + I (g, I (h + g I (h + I (g wheever the algebraic operatios mae sese; (9 sup I (f = I ( sup f for ay icreasig sequece (f for which I (f 1 > ; (10 if I ( (f = I if I (f 1 <. f for ay decreasig sequece (f for which
3 Daiell ad Riema itegrability 419 It ca be show that the set L 1 (I defied as L 1 (I = {f : X R I (f = I (f ± } is a real liear vector space of fuctios o X, such that (11 C L 1 (I, f g, f g belog to L 1 (I if f, g L 1 (I; (12 for ay sequece (f of L 1 (I, domiated i L 1 (I, i.e., f g for all N for some fuctio g L 1 (I, we have sup f L 1 (I, if f L 1 (I. Moreover, if the above sequece is poitwise coverget to a fuctio f, the we have f L 1 (I ad I (f = lim I (f, lim I ( f f = 0. If we deote by M the set of all subsets A of X such that the characteristic fuctio 1 A of A belogs to L 1 (I, the M is a σ-algebra o X, the map µ : M R + defied as µ (A = I (1 A = I (1 A is a measure o M, ay elemet f L 1 (I is M-measurable ad µ-itegrable. Moreover, we have I (f = fdµ ad, i particular, the above equality holds for ay f C. The elemets of M are geerally called Lebesgue measurable (w.r. to I (or Daiell-measurable w.r. to I. 2. RIEMANN INTEGRABILITY AND RIEMANN MEASURABILITY I the sequel we develop a theory close to the Riema theory of itegratio o the real lie with respect to the Lebesgue measure o R. The startig poit is a poitwise vector lattice of real fuctios C o a set X which cotais the real costat fuctios, ad a Daiell itegral I : C R. Defiitio 2.1. A real bouded fuctio h : X R is called Riema itegrable with respect to I or, simply, Riema itegrable if we have sup{i(f f C, f f} = if{i(f f C, f f }. We shall deote by R 1 (I the set of all Riema itegrable fuctios f : X R. For ay f R 1 (I, deote I(f = sup{i(f f C, f f} = if{i(f f C, f f }. Propositio 2.2 (Darboux criterio. A fuctio f : X R is Riema itegrable iff for ay ε > 0 there exist f, f i C (or f, f R 1 (I such that f f f ad I(f I(f < ε. Proof. The assertio follows directly from the above defiitio. Propositio 2.3. a The set R 1 (I is a poitwise vector lattice such that C R 1 (I L 1 (I ad for ay sequece (h from R 1 (I which decreases to zero, i.e., h = 0, the sequece (I (h decreases to zero.
420 Ileaa Bucur 4 b If (h is a sequece from R 1 (I, uiformly coverget to a fuctio h, we have h R 1 (I ad lim I (h = I(h, lim I ( h h = 0. Proof. a The iclusios C R 1 (I L 1 (I follow from the defiitios, o accout of the fact that C C i, C C s. Let h be a elemet of R 1 (I ad for ay ε > 0 let f, f C be such that f h f ad I(f f < ε. We cosequetly have f 0 h 0 f 0, (f 0 (f 0 f f, I(f 0 I(f 0 I(f I(f < ε. Sice the fuctios f 0 ad f 0 belog to C, we deduce that the fuctio h + = h 0 belogs to R 1 (I. Hece R 1 (I is a poitwise lattice. If the sequece (h from R 1 (I decreases to zero, by the above property (12 of the Daiell itegral, we have lim I (h = lim I (h = 0. Nevertheless, we ca directly prove this property. For ay ε > 0 ad for ay N choose f, f C such that f h f ad I (f f < ε/2. Cosequetly, for ay atural iteger we have =1 f h =1 f, ( I f f =1 =1 ( Sice the sequece ( we have lim I f = 0. =1 =1 =1 f =1 f =1 f =1 ( O the other had, the sequece f =1 f for ay N. Therefore, ( ( I (h I f ε + I =1 =1 I ( f f ε. ( f f, from C decreases to zero =1 f ( if h = 0, from C decreases ad h, lim I (h ε. The umber ε > 0 beig arbitrary, we get lim I (h = 0. b Let ε > 0 be arbitrary ad let 0 N be such that h h < ε for 0. We have h 0 ± ε R 1 ad h 0 ε h h 0 + ε, I (h 0 + ε I (h 0 ε = 2εI (1 X, i.e., h R 1 (I. Moreover, from the above cosideratios we deduce the relatios I ( h h εi (1 X, I (h I(h I ( h h ε, 0,
5 Daiell ad Riema itegrability 421 i.e., lim I ( h h = 0, lim I (h = I(h. Let us deote by R the set of subsets of X defied as R = {A A X, 1 A R 1 (I}. We refer to a elemet of R as Jorda measurable with respect to I. Obviously, we have R M. Propositio 2.4. The set R is a algebra of subsets of X. Proof. The assertio follows from the fact that R 1 (I is a poitwise vector lattice usig the equatios 1 A B = 1 A 1 B, 1 X\A = 1 1 A, A, B X. Theorem 2.5. If f : X R is a bouded fuctio, the the followig assertios are equivalet: 1 f is Riema itegrable w.r. to I. 2 There exists a icreasig sequece of R-step fuctios which coverges uiformly to f. 3 There exists a decreasig sequece of R-step fuctios which coverges uiformly to f. 4 For ay ε > 0 there exists a R-partitio ε = (A 1, A 2,..., A, A i R, A i = X, A i A j = φ if i j i=1 such that S (f, ε s (f, ε < ε, where S (f, ε, respectively s (f, ε, is the upper, respectively the lower, Darboux sum associated with f ad ε. Proof. The implicatios 2 4, 3 4 are obvious. 4 1. For ay atural iteger, 0, we cosider a partitio = (A 1, A 2,..., A of X, with A i R for ay i, such that i=1 M i µ (A i i=1 m i µ (A i < 1, sup f(x = M i, x A i if f(x = m i. x A i Sice 1 Ai R 1, we deduce that the fuctios ϕ, Ψ defied as ϕ = i m i1 Ai, Ψ = i M i1 Ai belog to R 1 (I, ad, moreover, we have ϕ f Ψ, I(ϕ = i m i µ(a i, I(Ψ = i M i µ(a i, I(Ψ I(ϕ < 1. Now, usig Propositio 2.2, we deduce that the fuctio f is Riema itegrable. 1 2 Without loss of geerality, we may assume that f 0. Sice for ay real umber r the set [f = r] belogs to M, the set D of real umbers defied as D = {r R µ ([f = r] > 0} is at most coutable. Let us ow show that the set [f > r] belogs to R for ay r R, r / D. Ideed, we have
422 Ileaa Bucur 6 1 [f>r] = sup ϕ, 1 [f r] = if Ψ, where, for ay atural iteger, the fuctios ϕ, Ψ are defied as ϕ = 1 (f r +, Ψ = 1 1 (r f +. Obviously, ϕ R 1 (I, Ψ R 1 (I, the sequece (ϕ is icreasig, the sequece (Ψ is decreasig ad the sequece (Ψ ϕ decreases to the fuctio 1 [f=r]. Sice µ ([f = r] = 0 we have lim I (Ψ ϕ = lim (Ψ ϕ dµ = µ ([f = r] = 0 ad, therefore, by Propositio 2.2 agai, ay fuctio g : X R, such that sup ϕ g if Ψ, belogs R 1 (I. I particular, the fuctios 1 [f>r], 1 [f r] belog to R 1 (I. Sice D is at most coutable, there exists a real umber a such that 0 < a 1 ad r a / D for ay ratioal umber r. For ay atural iteger 0 we cosider the fuctio ϕ defid as ϕ = a 2 2 p 1 =1 1 [f> a 2 ], where p N is such that f(x pa for ay x X. Obviously ϕ is a R-step fuctio, we have ϕ f ϕ + a 2, ad the sequece (ϕ is uiformly icreasig to f. Now, replacig f by f, we deduce that 2 3. Remar 2.6. I the case where X = [a, b] R ad C = {f : [a, b] R f cotiuous}, Theorem 2.5 geeralizes the famous Lebesgue criterio of Riema itegrability. Acowledgemets. The author would lie to express her thas to Professor Solomo Marcus for his valuable referece idicatios. REFERENCES [1] N. Boboc, Aaliză matematică. Ed. Uiv. Bucureşti, 1999. [2] Gh. Bucur, Aaliză matematică. Ed. Uiv. Bucureşti, 2007. [3] S. Marcus, La mesure de Jorda et l itégrale de Riema das u espace mésuré topologique. Acta Sci. Math. (Szeged 20 (1959, 2-3, 156 163. [4] M. Nicolescu, Aaliză matematică II. Ed. Acad. Româe, 1953. Received 20 Jauary 2009 Techical Uiversity of Civil Egieerig Departmet of Mathematics ad Computer Sciece Bd. Lacul Tei r. 124 020396 Bucharest, Romaia bucurileaa@yahoo.com