Finitely additive integration and local integration with respect to upper integrals

FINITELY ADDITIVE INTEGRATION AND LOCAL INTEGRATION WITH RESPECT TO UPPER INTEGRALS E. DE AMO1, R. DEL CAMPO1 and M. D´IAZ CARRILLO2 1 Dpto. de ´Algebra y An´alisis Matem´atico, Universidad de Almer´ıa, 04120 Almer´ıa, Spain e-mail: [email protected] and rcamp[email protected] 2 Dpto. de An´alisis Matem´atico, Universidad de Granada, 18071 Granada, Spain e-mail: [email protected] Abstract. We study the integration theory for general integral metrics when restricted to upper integrals q, finding improvements in the relation between the classes of the q-integrable and the ql-integrable functions. We give new results and notions which lead to the desirable characterizations of q-integrable functions as ql-integrable f with q(|f|) < ∞, and of ql-integrable functions via the integrability of their upper truncations, under natural conditions which are fulfilled in most finitely additive integration theories. 1. Introduction It is well known the general integration theory for integral metrics q, developed by Aumann and Sch¨afke in [1] and [11]–[14]. With the aid of this theory, D´ıaz Carrillo and G¨unzler gave in [6] an extension process for an arbitrary Loomis system (X, B, I), without monotone continuity assumptions on the elementary integral I, and proved convergence theorems using a suitable local mean convergence which can be traced back to Loomis in [10]. Key words and phrases: finitely additive integration, integral metric, upper integral, strong measurability. 2000 Mathematics Subject Classification: 28C05. − Moreover, a unified treatment of proper Riemann, Riemann-µ, abstract Riemann–Loomis, Daniell and Bourbaki integrals was given (doing proper choices of q). For a recent account of extension procedures we refer the reader to [4]. However, in many of such applications q is in fact an upper integral. Thus, we wish to investigate the benefits that can be obtained in general theory if we start from upper integrals instead of integral metrics. Section 2 presents some preliminaries where we set up notation and terminology to be used throughout the paper. In Section 3 we collect the most important results in integration theory with respect to upper integrals, giving natural conditions under which the q-integrable functions can be described as the class of q-regular functions. Section 4 is devoted to the study of local integration. A partial subadditivity for the localized upper integral ql is obtained (Proposition 4.6) and it is used to prove that ql(f) 5 (ql)∗(f) for all f ql-integrable (under regularity) in Theorem 4.10. In Section 5 we introduce and study the notions of upper and lower strong measurability. These new concepts will prove extremely useful to our purposes. In Section 6 our main results (Corollary 6.2 and Theorem 6.5) are stated and proved. The basic idea is to apply the results of Sections 3 and 4 in order to relate strong measurability for q and ql (see Theorem 6.1). As an application of the developed theory we conclude in Section 7 by giving a unified treatment of the finitely additive integral extension theories. It makes evident that our viewpoint sheds some new light on the way of obtaining general proofs for facts which were known only in special cases. 2. General framework. Preliminaries We extend the addition + on R to +, +˙ and .+ on R by a + b := 0, a +˙ b := ∞, a +. b := −∞, if a = −b ∈ {−∞, ∞}. We also note a − b := a + (−b), a−˙ b := a +˙ (−b), a . b := a +. (−b), ∀a, b ∈ R. The laws +, +˙ and .+ are commutative, + is distributive with 0 · (±∞) := 0, but not associative, and ˙+ is associative. The order 5 and absolute value | · | can also be extended to R by an obvious way and, setting a ∧ b := inf {a, b}, a ∨ b := sup {a, b}, a+ := a ∨ 0, a− := (−a) ∨ 0, ∀a, b ∈ R we still have the Birkhoff inequalities |a ∧ c − b ∧ c| 5 |a − b|, |a ∨ c − b ∨ c| 5 |a − b|, ∀a, b, c ∈ R and these other properties which will be used without a further explicit reference: a5|a|,|a|5c⇔ −c5a5c, ∀a, b, c ∈R,c=0 a5b˙ + (a−b),∀a, b ∈R a5b˙ +c⇒a−b5c, ∀a, b, c ∈R,c=0 (a−b)∧c=£a∧(c+b)¤−b, ∀a, b, c ∈]−∞,∞], c =0 (a˙ +b)∧c5£a∧(c−b)¤˙ +b, ∀a, b, c ∈R ¯ ¯|a|− |b|¯ ¯5|a−b|5|a−c|+|c−b|,∀a, b, c ∈R. For a nonempty set X,RXdenotes the class of all R-valued functions defined on X. All operations and relations between functions are defined pointwise. Given MjRX,let +M:= {f∈M:f=0}.Mis said to be a vector lattice if it is a real linear space under pointwise +, α·, such that f∧g, f∨g∈Mfor all f, g ∈M; then f∈Mimplies |f| ∈ M. For each ∅ 6=AjR,inf A, sup A∈R,where we use inf ∅:= +∞and sup ∅=−∞. 3. Integration with respect to upper integrals 3.1. Upper integrals. We begin by introducing the abstract concept of upper integral, which will be the basic tool in all this paper. Definition 3.1. A mapping q:RX→Ris said to be an upper integral if (1) q(0) = 0, (2) q(f˙ +g)5q(f)˙ +q(g)∀f, g ∈RX, (3) q(f)5q(g)∀f, g ∈RXwith f5g. We also recall the notion of integral metric in the sense of Sch¨afke. Actually, this is a simplified version of Sch¨afke’s definition [14, p. 120]. Definition 3.2. A mapping q: +RX→+Ris said to be an integral metric if (1) q(0) = 0, (2) q(f)5q(g) + q(h)∀f, g, h ∈+RX,f5g+h. We extend this definition to a mapping q:RX→Rsaying that qis an integral metric if q|+RXis an integral metric as above. It is clear that if qis an upper integral then qis an integral metric, and for each integral metric q,dq:RX×RX→Rgiven by dq(f, g) := q¡|f−g|¢ is a distance on RX,so one can do the next Definition 3.3. A function f∈RXis called q-M-integrable if it belongs to the closure of Min RXwith respect to the distance dq, i.e. ∀ε > 0,∃h∈M:q¡|f−h|¢< ε. The class of all q-M-integrable functions is denoted by Mq. Lemma 3.4. If qis an integral metric, 0∈MjRXand Mis closed with respect to ∗, being ∗ ∈ {+,−,∧,∨,| · |, α ·with α∈R}, then Mqis also closed with respect to ∗. We give now the basic property of the upper integrals and its principal consequences. Proposition 3.5. If qis an upper integral then ¯ ¯q(f)−q(g)¯ ¯5q¡|f−g|¢,∀f, g ∈RX. Proof. From f5g˙ + (f−g) we deduce that q(f)5q(g)˙ +q(f−g)5q(g)˙ +q¡|f−g|¢ and so q(f)−q(g)5q¡|f−g|¢. Interchanging fand ggives q(g)−q(f) 5q¡|g−f|¢. Hence −q¡|f−g|¢5q(f)−q(g)5q¡|g−f|¢, i.e. ¯ ¯q(f)−q(g)¯ ¯5q¡|f−g|¢.¤ Remark 3.6. Notice that all properties (1)–(3) of Definition 3.1 have been used to prove this one. Corollary 3.7. Every upper integral qis q-continuous on RX,i.e. q¡|f−fn|¢→0⇒q(fn)→q(f),∀f, fn∈RX. Corollary 3.8. Let be qan upper integral finite on MjRX. Then q is finite on Mq. Moreover, for each f∈Mq, and each sequence {hn}in M such as q¡|f−hn|¢→0(we say that {hn}defines fin Mq) we have q(f) = lim n→∞ q(hn)∈R. 3.2. Regularity and bidetermination. We are now concerned with additional conditions on q (fulfilled in most applications) yielding good properties on integration. Definition 3.9. Given q:RX→R,we define q∗by q∗(f) := −q(−f),∀f∈RX. If qis an upper integral then q∗is called the lower integral of q. Lemma 3.10. If qis an upper integral, then q∗satisfies: (i) q∗5qon RX, (ii) q∗(f+ .g)=q∗(f)+ .q∗(g)∀f, g ∈RX, (iii) q∗(f)5q∗(g)∀f, g ∈RXwith f5g. Proof. From 0 = q(0) = q¡f+ (−f)¢5q¡f˙ + (−f)¢5q(f)˙ +q(−f) one deduces that q∗(f)5q(f). The rest is left to the reader. ¤ Remark 3.11. Notice that, once again, it is needed to use all properties (1)–(3) of Definition 3.1 to prove (i). Definition 3.12. Let q:RX→Rand f∈RX.We say that fis qregular if q∗(f) = q(f)∈R. The class of all q-regular functions on Xis denoted by Reg (q), i.e. Reg (q) := ©f∈RX:q∗(f)=q(f)∈Rª. It follows directly from the properties of qand q∗(Definition 3.1 and Lemma 3.10) that the class Reg (q) has a good behaviour with respect to addition ˙ + when qis an upper integral; exactly: Proposition 3.13. If qis an upper integral and f, g ∈Reg (q)then f˙ + g∈Reg (q)and q(f˙ +g) = q(f) + q(g). Definition 3.14. Given MjRXand q:RX→R,we say that qis M- regular if q(h) = q∗(h)∈R,∀h∈M, i.e. if MjReg (q). The regularity of qon Mcan be extended to Mqfor every upper integral, as shown by the next proposition. Proposition 3.15. If qis an M-regular upper integral then qis Mqregular; that is, MqjReg (q). Proof. Let g∈Mq. There exists {hn}in Msuch that {hn}defines g in Mq, and so q(¯ ¯(−f)−(−hn)¯ ¯)→0. Therefore, using Corollaries 3.7 and 3.8 and the M-regularity of q, we have q∗(g) = −lim n→∞ q(−hn) = lim n→∞ q∗(hn) = lim n→∞ q(hn) = q(g)∈R.¤ Simple examples (for instance Example 7.5 below) make evident that this proposition becomes false without M-regularity. Combining the two previous propositions we obtain the ˙ +-additivity of qon Mq. Corollary 3.16. If qis an M-regular upper integral and f, g ∈Mq, then f˙ +g∈Mqand q(f˙ +g) = q(f) + q(g). The opposite relation between the classes Mqand Reg (q) depends on the following property of q. Definition 3.17. q:RX→Ris said to be determined by MjRXif q(f)= inf ©q(h) : f5h∈Mª,∀f∈RX. Proposition 3.18. If qis determined by Mand f∈RXwith q(f)= q∗(f)∈Rthen f∈Mq. Proof. Since q(f) = inf ©q(h) : f5h∈Mª∈R, given ε > 0, there exists h∈M,f5hsuch that q(f) + ε > q(h), and we have q¡|f−h|¢=q(h−f)5q(h) + q(−f) = q(h)−q∗(f) = q(h)−q(f)< ε which gives f∈Mq.¤ Example 7.6 shows that this is no longer true if the determination by M is dropped. Therefore, for M-regular upper integrals qdetermined by M, the classical characterization of q-M-integrability in terms of the equality of upper and lower integrals, follows by combining the preceding results, i.e. Corollary 3.19. If qis an M-regular upper integral determined by M then Mq=©f∈RX:q∗(f)=q(f)∈Rª. We have generalized this result to the case of an upper integral determined by a class M0bigger than the class Mwhere it is regular. Definition 3.20. Given M, M0jRXwith MjM0and q:RX→R, qis said to be bidetermined by (M, M0) if qis determined by M0,q∗|−M0is determined by Mand q > −∞ on M0. Note that, under M-regularity and with −MjM, to be bidetermined by (M, M) is equivalent to be determined by M. Theorem 3.21. If qis an M-regular upper integral bidetermined by (M, M0)and −MjMthen Mq=©f∈RX:q∗(f)=q(f)∈Rª. Proof. MqjReg (q) by Proposition 3.15. To prove the opposite inclusion, let f∈RXwith q∗(f)=q(f)∈Rand let ε > 0. From q(f) = inf ©q(h) : f5h∈M0ª∈R, we deduce that there exists h∈M0,f5hsuch that q(h) < q(f) + ε 2.Since q∗|−M0is determined by Mand −MjM, an easy computation shows that q(h) = sup ©q(u) : u5h, u ∈Mª. Clearly q(h)∈Rand consequently there exists u∈M,u5hsuch that q(h)−ε 2<q(u). Thus, we have q¡|f−u|¢5q¡|f−h|+|h−u|¢5q¡|f−h|¢+q¡|h−u|¢ 5q(h−f) + q(h−u)5q(h)−q∗(f) + q(h)−q∗(u) =q(h)−q(f) + q(h)−q(u)<ε 2+ε 2=ε. ¤ Taking M0=Mwith −MjMwe reobtain Corollary 3.19. This generalized version will be needed for some applications. Thus, regularity and bidetermination turn out to be natural conditions for the integration with respect to upper integrals, because under such conditions integrability can be decided by an intrinsic criterion for this type of functionals: to check that upper and lower integrals are equal and finite. 4. Local integration with respect to upper integrals Definition 4.1. Let qbe a functional defined on RX.We define the functional qlon RXby the formula ql(f) := sup ©q(f∧h) : h∈+Mª,∀f∈RX and, when qis an upper integral, qlis called the localized integral of q. Lemma 4.2. If qis an upper integral then qlis an integral metric. Proof. We only note that the inequality (f+g)∧h5f∧h+g∧h is valid for f, g ∈+RXand h∈+M.¤ Definition 4.3. The class of all ql-M-integrable functions with respect to the distance dql(f, g) := ql¡|f−g|¢on RXis denoted by Mql. We collect some elementary properties of qlthat we will use without explicit reference. Proposition 4.4. If qis a nondecreasing functional, then (i) ql5q. In particular, MqjMql. (ii) If ∃h∈+M,f5hthen ql(f) = q(f). (iii) If |M|jMand ∃h∈M,f5hthen ql(f) = q(f). It is clear that q∗∗ =qand qll =ql, but we can also consider the functionals given by iterated applications of the operations “∗” and “l” on q: ql∗, q∗l, q∗l∗, . . . . For instance, ql∗(f) := (ql)∗(f) = −ql(−f) = inf {q∗¡f∨(−k)¢:k∈+M},∀f∈RX. Proposition 4.5. Let qbe an upper integral. The general relations among the functionals q,ql,q∗,ql∗,q∗land q∗l∗are given by the following diagram (arrows mean 5): q∗l∗ q (40)== { { { { { { { {ql∗ (30) bbD D D D D D D D ql (2) OO q∗ (1) hhQQQQQQQQQQQQQQQQ (20) OO q∗l (3) aaC C C C C C C C(4) << z z z z z z z z Proof. (1) and (2) are already known. (3) follows from q∗(f∧h)5 q(f∧h), ∀h∈+Mand (4) from q∗(f∧h)5q∗(f), ∀h∈+M, both taking suprema. (20), (30) and (40) are obtained by applying ∗to (2), (3) and (4) respectively. ¤ We cannot expect as good behavior to local integration as to integration with respect to upper integrals, because the process of localization does not preserve upper integrals. However, we have proved that localization of upper integrals preserves some amount of additivity, which will be the key to find the relationship between the classes Mqand Mql; exactly: Proposition 4.6. Let qbe an upper integral and Mbe a vector lattice. Then ql(f˙ +h)5ql(f)˙ +ql(h),∀f∈RX,∀h∈M. Proof. If ql(f˙ +h) = −∞ then the inequality is trivial. If ql(f˙ +h)∈Rthen, given ε > 0, there exists k∈+Msuch that ql(f˙ +h)−ε5q¡(f˙ +h)∧k¢5q(f∧(k−h)˙ +h) + + + 5 q¡f ∧ (k − h)¢ ˙ q(h) 5 q¡f ∧ |k − h|¢ ˙ q(h) 5 ql(f) ˙ ql(h). Since ε > 0 is arbitrary, it follows that ql(f˙ +h)5ql(f)˙ +ql(h). Finally, if ql(f˙ +h) = ∞then, given α∈Rthere exists k∈+Msuch that α5q¡(f˙ + h)∧k¢and, as above, we have α5ql(f)˙ +ql(h) so ql(f)˙ +ql(h) = ∞.¤ Corollary 4.7. Let Mbe a vector lattice and qbe an M-regular upper integral. Then ql(f˙ +h) = ql(f) + ql(h),∀f∈RX,∀h∈M. Proof. The reverse inequality uses the one just proved: ql(f)5ql(¡f−(−h)¢˙ + (−h))5ql(f+h)˙ +ql(−h) 5ql(f+h)˙ +q(−h)5ql(f+h)−q(h) where q(h) = −q(−h)∈R.¤ As for the partial subadditivity of qllet us consider the following property, which is the best approximation to Proposition 3.5 that we get for ql: Corollary 4.8. If qis an upper integral and Mis a vector lattice then ql(f)−q(h)5ql¡|f−h|¢,∀f∈RX,∀h∈M. Proof. It follows immediately from ql(f)5ql¡(f−h)˙ +h¢5ql(f−h)˙ +ql(h)5ql¡|f−h|¢˙ +q(h).¤ We will now use this property to prove that ql(f)5(ql)∗(f) for all fin Mqlwhen qis an M-regular upper integral. The following lemma is needed. Lemma 4.9. Let qbe an upper integral. Assume that Mis a vector lattice, qis finite on M,f∈Mqland {hn}defines fin Mql. Then ©q(hn)ªis a Cauchy sequence in R. Proof. Since {hn}defines f, given ε > 0 there exists n0∈Nsuch that ∀n=n0, ql¡|f−hn|¢< ε. Thus, for n, m =n0we have, by Proposition 3.5 ¯ ¯q(hn)−q(hm)¯ ¯5q¡|hn−hm|¢=ql¡|hn−hm|¢ 5ql¡|hn−f|¢+ql¡|f−hn|¢< ε. ¤ 7. Applications and examples We now present a standard procedure to construct upper integrals from functionals on an MjRX. Definition 7.1. For ∅ 6=MjPjRXand T:P→Rwe define TM: RX→Rby TM(f) := inf ©T(h) : f5h∈Mª,∀f∈RX. With a slightly modified version of Lemma 11 in [6] and some related results, a unified treatment of the finitely additive integral extension theories can still be obtained. Lemma 7.2. Let MjRXwith 0∈Mand M˙ +MjM.If T:M→R is nondecreasing, T(0) = 0, and T(u˙ +v)5T(u)˙ +T(v),∀u, v ∈M, then TMis an upper integral, TM=Ton M, and (TM)M=TMon RX. Therefore, TMis an upper integral determined by Mwhich extends T. Moreover, TMis M-regular under closely related conditions: Lemma 7.3. Let qbe an upper integral and MjRXwith −MjM. The following are equivalent: (i) qis M-regular. (ii) qis +-additive and finite on M. Proof. (i) ⇒(ii). By Corollaries 3.16 and 3.8. (ii) ⇒(i). 0 = q¡h+ (−h)¢=q(h) + q(−h)⇒q(h) = −q(−h) = q∗(h) ∈R.¤ Corollary 7.4. Let MjRXbe a vector space. Assume T:M→Ris nondecreasing, T(0) = 0, and additive for +, i.e. T(u+v) = T(u) + T(v),∀u, v ∈M. Then TMis an M-regular upper integral determined by Mwhich extends T. A. Proper Riemann Integration. Given (X, Ω, µ), where Ω is a semiring of sets from Xand µ: Ω →[0,∞[ is additive, let us consider BΩ:= S(Ω,R), the step functions, and Iµ:= R·dµ on Ω, [9]. With q=I− µ:= (Iµ)BΩ, Corollary 7.4 says that qis a BΩ-regular upper integral determined by BΩand so Corollary 3.19 yields (BΩ)I− µ=©f∈RX:I− µ(f)=I+ µ(f)∈Rª=R1 e(µ, R) [9]. B. Abstract Riemann-µ-integration. With B:= BΩ,I:= Iµ,q=I− µthe BΩ-regular upper integral determined by BΩof A, and R1(µ, R):= Bql it follows that R1(µ, R)∩RX=R1(µ, R)([9, p. 70–144]) kL(X, Ω, µ, R) ([8, p. 112]), the latter inclusion being an equality if X∈Ω. Corollary 6.2 and Theorem 6.5 say that R1 e(µ, R) = {f∈R1(µ, R) : I− µ¡|f|¢<∞}, R1(µ, R) = {f∈RX:f±∧h∈R1 e(µ, R),∀h∈+B, I+ µ¡|f|¢<∞}. C. Loomis completions. Given a Loomis system (X, B, I) (BjRXvector lattice, I:B→Rlinear and positive) and q=I−:= IB, Corollary 7.4 guarantees that qis a B-regular upper integral determined by Band therefore Theorem 3.19 gives Bq∩RX=R1 etwo-sided completion Rof Loomis [10, p. 170] =Rprop(B, I) [7]. Moreover, it is known that BqlkR1 e(one-sided completion Uof Loomis [10, p. 178]), Bql=R1(B, I) (I-integrable functions of [7, p. 147]). A and B above are special cases. Corollary 6.2 and Theorem 6.5 here give the familiar descriptions in [7, Corollaries 1.7, 1.8] Rprop(B, I) = {f∈R1(B, I) : I−¡|f|¢<∞}, R1(B, I) = {f∈RX:f±∧h∈Rprop(B, I),∀h∈+B, I+¡|f|¢<∞} but nothing is said about the classes Bq ∧and Bq ∨in this case. These results are obtained by other methods (using local convergence). D. Finitely-additive Daniell extension. For a Loomis system (X, B, I), the following are introduced in [2]: I+(f) := sup ©I(h) : h5f, h ∈Bª, B+:= ©f∈RX:f=sup {h:f=h∈B}, I+(f)>−∞ª, B+:= ©f∈B+:I+(f+g)5I+(f) + I+(g),∀g∈B+ª. By Lemmas 7.2 and 7.3, q=I:= (I+)B+is a B-regular upper integral determined by B+. In fact, qis a B-regular upper integral bidetermined by (B, B+) and, therefore, applying Theorem 3.21, we still have B:= Bq=©f∈RX:I(f)=I(f)∈Rª, the summable functions B0of [2]. Moreover, Corollary 5.6 guarantees that Bq ∧=B∗ +=L∧,Bq ∨=B∗ −=L∨ of [2] and [3], respectively. Section 5 generalizes several properties, known for these classes B∗ +and B∗ −, to our general setting on upper integrals. E. Localization of the Daniell-analogue. For a Loomis system (X, B, I), q=Iof D leads to Bql=Lof [5]. We are able to apply all results in Section 6, yielding the new descriptions B={f∈L:I¡|f|¢<∞} and L={f∈RX:f±∧h∈B,∀h∈+B, I¡|f|¢<∞}. Since I5I−,one has BjLand Rprop(B, I)jR1(B, I)jL, with coinciding integrals. Example 7.5. For q:R→Rgiven by q(x) = xif x=0 and q(x) = 0 elsewhere, it is clear that {0}= Reg (q)⊂R=Rq. Example 7.6. Let X={1,2,3}and q:RX→Rdefined by q(x) := x1˙ + x2˙ +x3,∀x= (x1, x2, x3)∈RX=R3. It is clear that q(0) = 0, qis monotone and q(x˙ +y) = q(x)˙ +q(y) (since ˙ + is associative). Thus, qis an upper integral (it is even a Bourbakicontinuous integral norm). For Ω := ©∅,{1},{2,3}, Xª(a σ-algebra) and taking B:= BΩ, the step functions with respect to Ω (see A above), one has Bq=Bwhereas Reg (q) = RX. Obviously qis B-regular (see Lemma 7.3), but qis not determined by B. For instance, with x0:= (0,0,1), q(x0) = 1 <2 = q(0,1,1) = inf ©q(u) : x05u∈Bª. Moreover, qis not bidetermined by (B, B0) for any B0kB. If there exists an x0∈B0such that x06∈ Bthen x0(2) 6=x0(3) and therefore q∗(x0)6= inf ©q∗(u) : x05u∈Bª. References [1] G. Aumann, Integralerweiterungen mittels Normen, Arch. Math., 3(1952), 441–450. [2] P. Bobillo Guerrero and M. D´ıaz Carrillo, Summable and integrable functions with respect to any Loomis system, Archiv der Math., 49 (1987), 245–256. [3] P. Bobillo Guerrero and M. D´ıaz Carrillo, On the summability of certain µ-integrable functions, Archiv der Math., 52 (1989), 258–264. [4] M. D´ıaz Carrillo, Handbook of Measure Theory, Vol. I, Chap. 11 (2002), ed: E. Pap. [5] M. D´ıaz Carrillo and H. 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