DANIELL AND RIEMANN INTEGRABILITY

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1 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,

2 418 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 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.

4 420 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 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 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

6 422 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, [2] Gh. Bucur, Aaliză matematică. Ed. Uiv. Bucureşti, [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, [4] M. Nicolescu, Aaliză matematică II. Ed. Acad. Româe, Received 20 Jauary 2009 Techical Uiversity of Civil Egieerig Departmet of Mathematics ad Computer Sciece Bd. Lacul Tei r Bucharest, Romaia bucurileaa@yahoo.com