5 IntegrationWe will first discuss the question of integrability of bounded functions onclosed intervals, followed by the integrability of continuous functions (whichare nicer), and then move on to bounded functions with negligible disconti-nuities.The main tool will be to approximate the integral from above by theupper sum and from below by the lower sum, relative to various parti-tions. This method was introduced by the famous nineteenth century Ger-man mathematician Riemann, and it is customary to call these sums Rie-mann sums.5.1 Basic NotionsDefinition 5.1 If fis a bounded function on a closed interval [a, b], then thespan of f on [a, b] is given byspanf([a, b]) = supf([a, b]) − inff([a, b]),where supf([a, b]) (resp. inff([a, b])) denotes the supremum (resp. infimum)of the values of f on [a.b].If f is continuous on [a, b], then we know that it is bounded, and moreover,sup = max and inf = min (of f([a, b])).Definition 5.2 A partition of a closed interval [a, b ] is a collection of pointst0, t1, t2, . . . , tnsuch thatt0= a < t1< t2< ··· < tn−1< b = tn.Definition 5.3 A function S defined on [a, b] is called a step function ifthere is a partition P = {t0, . . . , tn} of [a, b], and constants c1, c2, . . . , cnsuchthat such thatS(x) = cjif x ∈ [tj−1, tj),and S(b) = cn.A proper definition of integration must allow such a (step) function tobe integrable, with its integral over [a, b], denoted∫baS, being the sum∑nj=1cj(tj− tj−1).1Definition 5.4 If P, P′are partitions of [a, b], we will say that P′is a re-finement of P iff the set of points in P is contained in the set of points ofP′.For example, P : t0= 0 < t1=12< t2== 1 and P′: t′0= 0 < t′1=14<t′2=12< t′3=34< t′4= 1 are both partitions of [0, 1], with P′a refinementof P .It is clear from the definition that given any two partitions P, P′of [a, b],we can find a third partition P′′which is simultaneously a refinement of Pand of P′. Such a P′′is called a common refinement of P, P′.Remark: The sum and the product of two step functions are also seento be step functions, by considering suitable refinements.Here is a quick definition of integrability:Definition 5.5 A bounded function f on [a, b] is integrable iff for everyε > 0, we can find a partition P = t0= a < t1< t2< ··· < tn−1< b = tnsuch that the sum∆f(P ) :=n∑j=1(tj− tj−1)spanf([tj−1, tj])is less than ε.Note that, for each ε > 0, the choice of P may depend on ε.The obvious question now is to ask if there are integrable functions. Onesuch example is given by the constant function f(x) = c, for all x ∈ [a, b].Then for any partition P = {a = t0< t1< . . . < tn= b}, the sup and inf off on each subinterval [tj−1, tj] coincide, making ∆f(P ) zero.Lemma 5.6 Every step function S is integrable on [a, b].Proof. Let S be the step functions associated to a partition P := t0=a < t1< t2< ··· < tn−1< b = tnand constants cj, so that S(x) = cjifx ∈ [tj−1, tj) and S(b) = cn. There is a jump in the value of S at each tjforj ∈ {1, 2, . . . , n − 1}. This is harmless and can be taken care of as follows.For any ε > 0, consider a refinement of P given byP′: a = t′0< t′1= t1−δ < t′2= t1< t′3= t2−δ < t′4= t2< ··· < t′2n−2= tn−1< t′2n−1= tn= b,2whereδ = ε/(n − 1)µ, with µ = max{c1, . . . , cn} − min{c1, . . . , cn}.Then, since the span of f is zero on each [t′2i− t′2i+1], we get∆f(P ) <n−1∑j=1δ(cj− cj−1) < ε,as each cj− cj−1< µ. 25.2 Upper and Lower SumsOne can interpret ∆f(P ) for each partition P of [a, b] as the difference be-tween certain upper and lower sums of Riemann.Definition 5.7 The upper, resp. lower, sum of f over [a, b] relative tothe partition P = {a = t0< t1< . . . < tn= b} is given byU(f, P ) =r∑j=1(tj− tj−1)supf([tj−1, tj])resp.L(f, P ) =r∑j=1(tj− tj−1)inff([tj−1, tj]),Of course we have, for all P ,L(f, P ) ≤ U(f, P ), and ∆f(P ) = U (f, P ) − L(f, P ).More importantly, it is clear from the definition that if P′is a refinementof P , thenL(f, P ) ≤ L(f, P′) and U(f, P′) ≤ U(f, P ).PutL(f) = {L(f, P ) | P partition of [a, b]} ⊆ RandU(f) = {U(f, P ) | P partition of [a, b]} ⊆ R.3Lemma 5.8 L(f) admits a sup, denoted I(f), called the lower integral off over [a, b]. Similarly, U(f) admits an inf, denoted I(f), called the upperintegral.Proof. Thanks to the discussion in Chapter 1, all we have to do isshow that L(f) (resp. U(f)) is bounded from above (resp. below). Sowe will be done if we show that given any two partitions P, P′of [a, b], wehave L(f, P ) ≤ U(f, P′), as then L(f) will have U(f, P′) as an upper boundand U(f) will have L(f, P ) as a lower bound. Choose a third partition P′′which refines P and P′simultaneously. Then we have L(f, P ) ≤ L(f, P′′) ≤U(f, P′′) ≤ U(f, P′). Done.2We always haveI(f) ≤ I(f).Lemma 5.9 A bounded function f is integrable over [a, b] iff I(f) = I(f).Proof Suppose f is integrable. Then, by definition, ∆f(P ) = U(f, P ) −L(f, P ) becomes arbitrarily small as P goes through a sequence of refine-ments. This forces the equality I(f) = I(f) in the limit. Conversely, giventhis equality, the difference U( f, P ) − L(f, P ) must become less than anygiven ε > 0, for a suitably refined P . 2When such an equality holds, we will simply write I(f) for I(f) (= I(f)),and call it the integral of f over [a, b].Quite often we will also writeI(f) =b∫af orb∫af(x) dx.In practice one is loathe to consider all partitions P of [a, b]. The followinglemma tells us something useful in this regard.Lemma 5.10 Let f be a bounded function on [a, b]. Suppose {Pn} is aninfinite sequence of partitions, with each Pnbeing a refinement of Pn−1, suchthat the corresponding sequences {U(f, Pn)} and {L(f, Pn)} both converge toa common limit λ in R. Then f is integrable with λ =b∫af(x)dx.4Note that for such a step function f defined by (P, {cj}), we have anexplicit formula for the integral, namelyb∫af(x)dx =n∑j=1cj(tj− tj−1).5.3 Integrability of monotone functionsLet f : A → R be a function, with A a subset of R. Recall that f ismonotone increasing (resp. monotone decreasing), iff we havex1, x2∈ A, x1< x2=⇒ f (x1) ≤ f(x2) (resp. f(x1) ≥ f(x2)).A monotone function f on a closed interval [a, b] is bounded on [a, b]. Thisis clear because f is bounded by f(a) on one side and by f(b) on the other.Theorem 5.11 Let f be a