Regulated Functions: Bourbaki's Alternative to the Riemann Integral Author(s): S. K. Berberian Reviewed work(s): Source: The American Mathematical Monthly, Vol. 86, No. 3 (Mar., 1979), pp. 208-211 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2321526 . Accessed: 27/12/2011 11:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org 208 S. K. BERBERIAN [March whichestablishes (3). Equalityholds ifAi= nsince 02(x) is strictlylog-convex. Now let 43(x) =(1- x)/x; then +3(x) is log-convexfor 0 <x < ' and log-concavefor 2 6?x< 1. Also let A= (A1,...,A), where0 <Ai < . Clearly As Ai ,.., i ) so that If(lA,)>[Al Agn p[n2/AJ establishing(4). Finally,let A1= ((n- l)A 1...X(n- l)An) and letA 2 = (l -A 1 -An) whereAi > 0,lAi 1. It can be easilyverified that A1>-A2. By Theorem1, g(x)=lli"=1xi is a Schur-concavefunction. It thenfollows that n n n (n-lI)n II Ai= II (n- I)Ai < (I(- A), i=l i=l i=l provinginequality (2). It is apparentthat many additionalinequalities of the Weierstrassproduct type can be formulatedand provedby choosingthe appropriatelog-concave function, forming products to obtaina Schur-convexfunction, and thenusing Definition 2 above. The researchfor this paper was sponsoredby AirForce Office of ScientificResearch, USAF, AFSC, under AFOSR Grant74-2581C. Reproduction is permitted for any purpose of theUnited States Government. References 1. M. S. Klamkin, D. J. Newman, Extensionsof the Weierstrassproduct inequalities, Math. Mag., 43 (1970) 137-140. 2. M. S. Klamkin,Extensions of the Weierstrassproduct inequalities II, thisMONThY, 82 (1975) 741-742. 3. A. Ostrowski,Sur quelques applicationsdes fonctionsconvexes et concaves au sens de I. Schur,J. Math. PiiresAppl., 31 (1952) 253-292. 4. D. S. Mitrinovic,Analytic Inequalities, Springer-Verlag, Heidelberg, 1970. DEPARTMENTOF STATISTICS,UNIvERsrIT OF KENTucKY, LExiNGTON, KY 40506. DEPARTMENTOF STATISTICS, FLORIDA STATE UNIVERSrrY,TALLAHASSEE, FL 32306. CLASSROOM NOTES EDITED BY DEBORAH TEPPER HAIMO AND FRANKLIN TEPPER HAIMO Materialfor this department should be sentto ProfessorDeborah Tepper Haimo, Department ofMathematics, UniversityofMissouri, St. Louis,MO 63121. REGULATED FUNCTIONS: BOURBAKIS ALTERNATIVE TO THE RIEMANN INTEGRAL S. K. BERBERLAN 1. Introduction.At theoutset, I hastento say thatI remaina "Riemannloyalist": pound for pound,the Riemannian circle of ideas can't be beat forits instructionalvalue to thebeginning 1979] CLASSROOM NOTES 209 studentof analysis.Consequently, I wouldn't go so faras to suggestthat the theory of regulated functionsreplace the Riemannintegral in the beginningundergraduate analysis course; how- ever, in a graduatecourse in real variables,the theoryof regulatedfunctions can be an entertainingalternative to a routinereview of the Riemannintegral; and it is in some waysa moreinstructive prelude to the Lebesguetheory, as I hope to persuadethe readerin thisbrief "comparativeanatomy" of integrationtheories. 2. Regulatedfunctions. In thefollowing, [a, b] denotesa fixed,nondegenerate closed interval of the real line R, and f, g, F,... are real-valuedfunctions on [a, b], assumedto be bounded whenthey need to be bounded. In theclass of Riemann-integrablefunctions, Exhibit A is theclass of continuousfunctions; ExhibitB is the class of monotonefunctions (and its linearspan, the functionsof bounded variation).There is a propertyshared by thesetwo special classes of functions:at everypoint of theinterval, the function possesses finite one-sided limits. Such a functionis said to be regulated. (A rationalefor the terminology:such a functionis "limitedon the left"and "limitedon the right."The classicalterm: function with only discontinuities of the "firstkind.") The regulated functionsform an algebraof functionsfor the pointwise operations, by thealgebraic properties of limitibility(in otherwords, by the continuityof the algebraicoperations on real numbers). Every regulatedfunction is bounded (by an easy contradictionargument based on the Weierstrass-Bolzanotheorem). The uniformlimit of regulatedfunctions is regulated,by the "iteratedlimits theorem" [5, p. 149,Th. 7.11]. From the viewpoint of integrationtheory, the most transparentexample of a regulatedfunction is a stepfunction, that is, a functionwith finitely many values, each assumed on an interval,possibly degenerate (in otherwords, a linear combinationof characteristicfunctions of intervals,possibly degenerate). Every uniform limit of step functionsis regulated.In fact,there are no otherregulated functions: every regulated functionis theuniform limit of stepfunctions (by essentiallythe same argument,based on the Heine-Boreltheorem, used to show thata continuousfunction is uniformlyapproximable by step functions)[1, Ch. 2, ? 1, no. 3, Th. 3]; [2, p. 139,Th. 7.6.1].A step functionf comesvery close to havinga continuousanti-derivative: there exists a continuousfunction F (necessarily piecewiselinear) whose derivative exists and is equal tof(x) exceptat thefinitely many points of discontinuityof f; suggestively, 3F'(x) = f(x) f.e., where"f.e." means "withfinitely many exceptions." This implies,by a standardtheorem on term-by-termdifferentiation [5, p. 152,Th. 7.17],that for every regulated function f, thereis a continuousfunction F whose derivativeexists and is equal to f(x) forall but countablymany values of x; suggestively, 3F'(x) =f(x) c.e., where"c.e." means"with countably many exceptions." 3. An elementaryintegration theory. The foregoingdiscussion suggests this definition: call F a primitiveof f if (1) F is continuous,and (2) 3F'(x) =f(x) c.e. (More precisely,one could say thatF is a "c.e.-primitive"of f. If F'(x) =f(x) forall x, one calls F a strictprimitive of f; we remarkthat the range off mustthen be an interval[5, p. 108,Th. 5.12].)As notedabove, every regulatedfunction has a primitive.(The converseis false; see Section6 below.) Iff has a primitiveF, one is temptedto define fbf = F(b)-F(a). This'is a legitimatedefinition, since any twoprimitives of f mustdiffer by a constant,by virtue of thefollowing result [1, Ch. 1, ?2, no. 3, Cor. of Th. 2]: If H is continuousand if 3H'(x)=0 c.e., thenH is constant.(Indeed, it sufficesto supposethat H is continuousand that3H,'(x) = 0 210 S. K. BERBERIAN [March c.e., whereHr' denotesright derivative.) One has herethe makingsof an elementarytheory of integration:call a functionf: [a, b]-*R integrableif it has a primitiveF in theabove sense,and thendefine f bf= F(b) - F(a). The integrablefunctions form a linearspace (but not a linear algebra,as notedin Section6); the uniformlimit of integrablefunctions is integrable,by the theoremon term-by-termdifferentiation mentioned earlier. In thisintegration theory, the class of "primitives"is the class of functionsF: [a,b]-*R such that(1) F is continuous,and (2) 3F'(x) c.e. Examples: F continuousand monotone;more generally,(i) F continuousand of bounded variation[6, p. 1071,or (ii) F continuousand convex[1, Ch. 1, ?4, no. 4, Prop. 8]. (Incidentally,if f is monotone(hence f has a primitive),then every primitive of f is convex[1, Ch. 2, ?1, no. 3, Prop. 4].) REMARK. Iff is integrableand g =f c.e., theng is integrable(with the same primitivesas I); thisfollows trivially from the definitions. 4. The Lebesgue integral.Lebesgue's "fundamentaltheorem of calculus" gives a succinct characterizationof Lebesgue-integrability[6,pp. 198,201]: A functionfis Lebesgue-integrableif and onlyif thereexists an absolutelycontinuous function F such thatF'(x) =f(x) a.e.; one then has f lf(x)dx = F(b) - F(a). Relevanthere is the followingfact: If H is absolutelycontinuous and H'(x)=0 a.e., thenH is constant[6, p. 205]. (Incidentally,one cannotweaken "absolutely continuous"to "continuousof boundedvariation," as is shownby Lebesgue'sfamous example of a nonconstantcontinuous increasing function whose derivativevanishes almost everywhere [3, p. 96].) In Lebesgue'stheory, the class of "primitives"is the class of absolutelycontinuous functions(such a functionalways possesses a derivativea.e.). REMARK. If f is Lebesgue-integrableand g =f a.e., theng is Lebesgue-integrable(with same "6absolutelycontinuous, a.e.-primitives" as J). 5. The Riemannintegral. Here f denotes a bounded function,Df its set of points of discontinuity.The succinctcriterion for Riemann-integrabilityis that of Lebesgue:f is Rie- mann-integrableif and only if Df is Lebesgue-negligible[6, p. 142].Then the formulaF(x)= fAf(t) dt definesan absolutelycontinuous (indeed, Lipschitz) function, with F'(x) =f(x) a.e. and in particularfor x z Df. The class of "primitives"F forthis theory is somewhatclumsy to describe:(1) F is Lipschitz (henceabsolutely continuous, with bounded derivative), and (2) thereexists a boundedfunction f: [a, b]--R, continuousa.e., such thatF'(x) =f(x) a.e. The Riemann-integrablefunctions form an algebraof functions,closed under uniform limits; thisis easy to see, forinstance, from Lebesgue's criterion. REMARK. Iff is Riemann-integrableand