Abstract Riemann integrability and measurability

ABSTRACT RIEMANN INTEGRABILITY AND MEASURABILITY E. de Amo, Almería, R. del Campo, Almería, and M. Díaz Carrillo, Granada Abstract. We prove that the spectral sets of any positive abstract Riemann integrable function are measurable but (at most) a countable amount of them. In addition, the integral of such a function can be computed as an improper classical Riemann integral of the measures of its spectral sets under some weak continuity conditions which in fact characterize the integral representation. Keywords: finitely additive integration, localized convergence, integral representation, weak continuity conditions, horizontal integration 1. Introduction Given a Loomis system (X, B,I)without any continuity conditions for the basic integral I, Díaz Carrillo and Mu˜noz Rivas introduced in [13] a general extension process using a suitable localized convergence, thus obtaining the class R1(B,I)of abstract Riemann integrable functions. This extension process subsumes μ-Riemann [18], Riemann-Loomis [22], and Dunford-Schwartz [15] integrations. The classical problem of the relation between the integrability of a function f and the measurability of its spectral sets, f−1(]r, +∞[),forr∈ Ê , has been widely studied by several authors in those measure-theoretic contexts (see [23] and [24], for instance). We now carry on this discussion in our functional setting and prove that every abstract integrable function is quasi-measurable, that is, all its spectral sets are measurable but at most a countable amount of them. Moreover, under some weak continuity conditions, we are able to obtain a formula that allows to reconstruct the integral of an integrable function through the measures of its spectral sets. Our results generalize the previous corresponding ones from the measure-theoretic point of view to the functional context in which the integral need not even be induced by any finitely additive measure. Section 2 is devoted to introducing notation and preliminary results to make the paper self-contained. In Section 3, the additional conditions we need are defined and their interactions with the localized convergence are presented. The measurability of the spectral sets of an integrable function is discussed in Section 4. The last section shows that, under stonian and lower and upper continuity conditions, it is possible to obtain the integral of f∈R1(B,I)by adding the measures of its spectral sets. In fact, these conditions characterize integral representation in this new situation (see [9], [17], [19] and [21]). 2. Preliminaries For Ê := Ê ∪{−∞,+∞},where Ê is the real line, we extend the usual addition in Ê to Ê by the conventions r+s:= 0 if r=−s∈{−∞,+∞} and r−s:= r+(−s). We also set a∨b:= max{a, b},a∧b:= min{a, b},a+:= a∨0and a−:= −(a∧0). Given an arbitrary nonempty set X,let Ê Xconsist of all functions defined on X with values in Ê . All operations and relations in Ê Xare defined pointwise, with the conventions inf ∅:= +∞and sup ∅:= −∞. A functional T: Ê X→ Ê will be called subadditive if T(f+g)⩽T(f)+T(g)for all f,g ∈ Ê Xunless T(f)=−T(g)=+∞and T(f)=−T(g)=−∞. The notion of asuperadditive functional is introduced in the completely dual way. Atriple(X,B, I)is called a Loomis system if Xis a nonempty set, B⊆ Ê Xis a vector lattice of real functions and I:B→ Ê is a positive (i.e., I(h)⩾0for all h∈Bwith h⩾0) linear functional. We set +B:= {h∈B:h⩾0}. Given (X,Ω,μ)with μa finitely additive measure and Ωa ring, we call (X, BΩ,I μ) the induced Loomis system,whereBΩis the vector lattice of μ-simple functions, BΩ:= h∈ Ê X:h= n  i=1 aiχAi,a i∈ Ê ,A i∈Ω,μ([h=0])<+∞, and Iμis its canonical elementary integral given by Iμ(h):= n  i=1 aiμ(Ai),∀h∈BΩ. 2.1. Proper Riemann integration. Following Loomis in [22] we extend the elementary functional I on B to the functionals I+ and I− (oscillation integrals) over the class Ê Xof extended real valued functions, I−(f):=inf{I(h): h∈B,h ⩾f}(upper functional of I), I+(f):=sup{I(h): h∈B,h ⩽f}(lower functional of I), which evidently verify I−(f)=−I+(−f), are positively homogeneous and monotone, I−is subadditive and I+is superadditive. We also consider the class of the properly Riemann integrable functions Rprop(B,I)={f∈ Ê X:I+(f)=I−(f)∈ Ê }, which is a vector lattice where the functional I:= I+=I−is linear and positive, i.e., it is an integral which extends the initial I. For every f∈ Ê X, the following statements are equivalent: (i) f∈Rprop(B,I). (ii) ∀ε>0∃h, g ∈Bsuch that h⩽f⩽gand I(g−h)<ε. (iii) ∀ε>0∃h∈Bsuch that I−(|f−h|)<ε. Hence, Rprop(B,I)is the closure of the vector lattice Bwith respect to the integral seminorm I−(|·|)(see [4] and [27]). Note that the particular case of a Loomis system induced by the semiring {[a, b[: −∞ <a⩽b<+∞} and the finitely additive measure μ([a, b[) = b−a, leads to the classical Riemann integrable functions (as in [4] and [18, p. 216]). 2.2. Abstract Riemann integration. Since for proper Riemann integration there are no satisfactory Lebesgue convergence type theorems to make a consistent integration theory, Díaz Carrillo and Mu˜noz Rivas introduced in [13] the class R1(B,I)of the abstract Riemann integrable functions as R1(B,I):={f∈ Ê X:∃{hn}in B,I-Cauchy; {hn}−→f(I−)} where {hn}I-Cauchy means that I(|hn−hm|)→0,forn, m →+∞,and {hn}−→f(I−)means that {I−(|fn−f|∧h)}→0,∀h∈+B.Thisnotionof local I-convergence allowed them to obtain convergence theorems for R1(B,I)(see theorems 2.3, 2.4 and 2.7 in [13]). Moreover, for each f∈R1(B,I)we set I(f) := lim n→+∞I(hn)for any I-Cauchy sequence {hn}in Bsuch that {hn}−→f(I−). The definition does not depend on the particular sequence {hn}and no confusion arises with this notation since Rprop(B,I)⊆R1(B,I)with coinciding integrals I. There are several useful characterizations for the class R1(B,I). On the one hand, we have R1(B,I)={f∈ Ê X:I+(|f|)<+∞,f±∧h∈Rprop,∀h∈+B}, which, in particular, says that +R1(B,I)={f∈ Ê X:I+(f)<+∞,f ∧h∈Rprop,∀h∈+B}; furthermore, I(f)=I+(f)for all f∈+R1(B,I). On the other hand, given f∈ Ê X,thelocalized functional I− lin the sense of Schäfke (see [28]) is defined as I− l(f):=sup{I−(f∧h): h∈+B}. It is easily verified that I− lis positively homogeneous, monotone and subadditive. Moreover, (I− l)l=I− l,I+⩽I− l⩽I−and I− l(f)=I−(f)if f⩽hfor some h∈+B. Theorem 2 in [14] guarantees that R1(B,I)is the closure of Bin Ê Xwith respect to the integral seminorm I− l(|·|),andI− l(f)=I(f), for all f∈R1(B,I)(Iis the only I− l-continuous extension of Ifrom Bto R1(B,I)). It is also possible to provide another description of the class R1(B,I)by means of the upper and lower essential functionals due to Anger and Portenier (see [3]), I•(f):= inf v∈+Bsup u∈+B I−((f∧u)∨v),∀f∈ Ê X, I•(f):=−I•(−f),∀f∈ Ê X. Evidently I•coincides with I− lon the positive functions and therefore, theorems 4.4 in [3] and 5 in [11] guarantee that R1(B,I)can be represented in the following way: R1(B,I)={f∈ Ê X:I•(f)=I•(f)∈ Ê }. Finally, we define the oscillation integrals for R1(B,I): ∀f∈ Ê X, I∗(f):=inf{I(g): g∈R1(B,I),g ⩾f},(1) I∗(f):=sup{I(g): g∈R1(B,I),g⩽f},(2) which verify that I∗(f)=−I∗(−f),both I∗ and I∗ are positively homogeneous and increasing, I∗ is subadditive and I∗ is superadditive, and both extend I on R1(B, I). In fact, in [2, Cor. 3.9] it is proved that the extension process for the initial Loomis system (X,R1(B,I),I)through oscillation integrals I∗and I∗is closed; i.e. R1(B,I)={f∈ Ê X:I∗(f)=I∗(f)∈ Ê }. Since B⊆R1(B,I)we have that I+⩽I∗⩽I∗⩽I−.Moreover,iff⩾0and there exists h∈Bsuch that f⩽h,thenI−(f)=I∗(f). 3. Additional conditions It is worth pointing out that abstract Riemann integration coincides with classical Daniell integration [8] when monotone continuity is assumed, but in order to obtain the previous integral extension process, we have not used any additional condition on the initial vector lattice B, nor on the linear functional Idefined on it, and hence it allows to subsume most of finitely additive integration theories. Nevertheless, weak continuity conditions (on the Loomis system) need to be introduced if we want to obtain the representation result we desire. Definition 3.1. The vector lattice Bis stonian if f∧1∈B, ∀f∈B(equivalently, f∧r∈B, ∀f∈B, ∀r∈ Ê ). Definition 3.2. ALoomissystem(X,B,I)is called C∞or upper continuous if lim r→+∞I∗(f−f∧r)=0,∀f∈+B, and it is called C0or lower continuous if lim r→0I∗(f∧r)=0,∀f∈+B. The stonian condition on Bis hereditary for the class R1(B,I),asissaidin Lemma 3.3. If Bis stonian, then R1(B,I)is stonian too. Proof. For f∈R1(B,I),thereexistsanI-Cauchy sequence {hn}in B,and {hn}→f(I−).Thus{hn∧1}→f∧1(I−),and{hn∧1}is I-Cauchy, because the inequality |hn∧1−hm∧1|⩽|hn−hm|is valid for all n, m ∈  . Therefore, f∧1∈R1(B,I); i.e., R1(B,I)is stonian.  We now study conditions under which C0and C∞are hereditary from Bto R1(B,I). We call attention to the fact that, while for C0the stonian condition is necessary for the initial vector lattice B,forC∞it is possible to obtain the heritage directly for the class R1(B,I). Lemma 3.4. For an arbitrary Loomis system (X,B,I)we have: (i) If Bis C0and stonian, then R1(B,I)is C0. (ii) If Bis C∞,thenR1(B,I)is C∞. Proof. (i) Let f∈+R1(B,I)and fn:= f∧1/n,∀n∈ Æ .SinceBis stonian so is R1(B,I). Therefore, fn∈R1(B,I),∀n∈ Æ . Clearly, {fn}converges uniformly to 0on Xand, since Bis C0, it is easy to check that {fn}→ 0(I−).Moreover, |fn|⩽f∈R1(B,I)for all n∈ Æ and hence the Dominated Convergence Theorem (see [13, theorem 2.7]), guarantees that I(fn)=I(f∧1/n)→I(0) = 0. Thus, given ε>0,thereexistsm∈ Æ such that I(fm)<ε, and hence, given 0<r<1/m, we have I(f∧r)⩽I(fM)⩽εwhich proves that R1(B,I)is C0. (ii) Let us consider f∈+R1(B,I)and ε>0.SinceI+(f)⩽I(f)<+∞,there exists h∈+Bwith h⩽fsuch that I(f)−1 4ε<I(h)and so I(f−h)<1 4ε. Moreover, since Bis C∞,thereexistss>0such that I∗(h−h∧r)<1 2ε, for all r⩾s. Then, given r⩾s,wehave I∗(f−f∧r)=I∗(|f−f∧r|)⩽I(f−h)+I∗(h−h∧r)+I∗(|h∧r−f∧r|) and using the Birkhoff inequalities we deduce that I∗(f−f∧r)⩽I(f−h)+I∗(h−h∧r)+I(f−h)<ε 4+ε 2+ε 4=ε.  Condition C∞on R1(B,I)means that it is possible to approximate the elements in R1(B,I)through their I−-local upper truncations with constants, i.e, if R1(B,I) is C∞,then{f∧rn}→f(I−)for all {rn}→+∞,with{rn}⊂ Ê +and for all f∈R1(B,I). In fact, under the additional condition that R1(B,I)is stonian, the converse is true by the Dominated Convergence Theorem, [13, Th. 2.7]. Analogously, C0condition on R1(B,I)says that {f∧rn}→0(I−)for all {rn}→0,with{rn}⊆ Ê +and for all f∈R1(B,I). Some examples of Loomis systems which are C∞but are not C0and viceversa can be found in [19]. 4. Measurability for abstract Riemann integration First of all, we must make precise the meaning of measurability in this functional context: Given an arbitrary Loomis system (X, B,I)we consider the finite measure space (X,Ω,μ)induced by R1(B,I),thatis, Ω:={A⊆X:χA∈R1(B,I)}and μ(A):=I(χA),∀A∈Ω. Therefore, the measurable sets will be those sets whose characteristic function is abstract Riemann integrable, and their measure will be the integral of this function. In addition, the spectral sets of a function are defined in the following way: Definition 4.1. Given f∈ Ê X,the spectral sets of fare the sets [f⩾r]:= {x∈X:f(x)⩾r}and [f>r]:={x∈X:f(x)>r},withr∈ Ê . With these definitions, the classical notion of measurability can be formulated in this functional context as those functions whose all spectral sets are measurable. Unfortunately, this kind of measurability does not have a good behavior with respect to Riemann-type integration since there are integrable functions (even classical Riemann integrable functions) such that they are not measurable in this sense (see [16]). Nonetheless, employing this slightly modified version of the classical notion of measurability we obtain some results in this direction: Definition 4.2. Given a ring Rin Xand f∈ Ê X,wesaythatfis quasi-Rmeasurable if there is a countable set Nin Ê such that [f⩾r],[f⩾r]∈Rfor all r∈ Ê \N. With this new terminology (inspired by Maharam [24]), Ridder proved in [26] that integrability and quasi-measurability are equivalent properties for the particular case of proper Riemann integration with respect to a finite finitely additive measure. In her paper, Maharam introduced an improper integration theory with respect to a finitely additive measure and extended this result to this new situation (see [24, Th. 5.1]). Finally, Luxemburg proved in [23, Th. 4.10] that integrable functions are quasi-measurable for Dunford-Schwartz integration. Using the previously mentioned description of abstract Riemann integration as the essential integration of Anger and Portenier and some techniques employed before in [5], we have now proved that, under the stonian condition, every integrable function is quasi-measurable, that is: Theorem 4.3. If Bis stonian, then every abstract Riemann integrable function is quasi-Ω-measurable. Proof. Letf∈R1(B,I)and let r0<r 1<r 2<...<r nbe real numbers. For each k∈{1,...,n},setfk:= (f∧rk−f∧rk−1)/(rk−rk−1),Ak:= [f⩾rk]and Bk:= [f>r k].SinceR1(B,I)is stonian we have fk∈R1(B,I),∀k∈{1,...,n}. Moreover, χAk⩽fk⩽χAk−1and χBk⩽fk⩽χBk−1,∀k∈{1,...,n}. Therefore we can restrict ourselves to proving the measurability of the Ak’s except for a countable amount of them (the reasoning is analogous for the Bk’s). From χAk⩽fk⩽χAk−1we deduce that I•(χAk),I •(χAk)∈ Ê ,∀k∈{1,...,n}, I•(χAk)−I•(χAk)⩽I(fk)−I(fk+1),∀k∈{1,...,n−1}, I•(χAn)−I•(χAn)⩽I(fn). Consequently, n  k=1 [I•(χAk)−I•(χAk)] ⩽ n−1  k=1 [I•(χAk)−I•(χAk)] + [I•(χAn)−I•(χAn)] ⩽ n−1  k=1 [I(fk)−I(fk+1)] + I(fn)=I(f1)−I(fn)+I(fn)=I(f1). Thus, we have obtained the inequality (3) n  k=1 [I•(χAk)−I•(χAk)] ⩽I(f1) Let Nbe the set of all real numbers rsuch that χAr∈ R1(B,I),whereAr:= [f⩾r]. It is clear that Ncan be written as N= a∈  ,ε∈  + {r>a:I•(χAr)−I•(χAr)⩾ε}. Assume that for some a∈ É and some ε∈ É +there exists an infinite number of r∈ Ê with r>aand I•(χAr)−I•(χAr)⩾ε.Thenforeachn∈ Æ we can select real numbers r2< ... < r namong them and therefore, applying inequality (3) (with r0=a−1and r1=a), we conclude that I(f1)⩾ n  k=2 [I•(χAkh)−I•(χAk)] ⩾ n  k=2 ε=(n−1)ε, ∀n∈ Æ . Letting n→+∞, we obtain that I(f1)=+∞, but this contradicts the fact that f1∈R1(B,I). We have proved that for each a∈ É and for each ε∈ É +there exists a finite number of real numbers rsuch that r>aand I•(χAr)−I•(χAr)⩾ε. Therefore N is countable and χAr∈R1(B,I)for all r∈ Ê \N. Theorem 4.3 says that if Bis stonian and f∈R1(B,I),thenthesetT(f):= {t∈ Ê :[f⩾t]∈Ω}is co-countable and hence dense. The converse of this theorem is still an open question, but at least with the aid of the Dominated Convergence Theorem for the class R1(B,I)([13, th. 2.7]) we have proved that, in this case, both the properties are equivalent for T(f), the same as it occurs for Maharam integration theory with respect to a finitely additive measure (see [24, cor. 5.2]). Proposition 4.4. Let f∈R1(B,I)and T(f)={t∈ Ê :[f⩾t]∈Ω}.The following statements are equivalent: (i) T(f)is co-countable. (ii) T(f)is dense in Ê . Proof. (i) ⇒(ii) is evident. Therefore assume that T(f)is dense and let us see that, in fact, it is co-countable. Consider the function F: Ê −→ Ê given by F(t):=I∗(χ[f⩾t])for all t∈ Ê ,which is, clearly, decreasing. Let C(F)be the set of all continuity points of F. It is well known that, because of the monotony of F,C(F)is co-countable (see, for example, [6]), so it is enough to show that C(F)⊆T(f),andthenT(f)will be a co-countable set. Let t0∈C(F).SinceT(f)is dense in Ê we are able to choose an increasing sequence {tn}→t0,withtn∈T(f), for all n∈ Æ and hence {F(tn)}→F(t0), i.e., {I∗(χ[f⩾tn])}→I∗(χ[f⩾t0]). Setting gn:= χ[f⩾tn]for all n∈ Æ ∪{0}, we have that the sequence {gn}is decreasing, I∗(gn)→I∗(g0)and gn∈R1(B,I)for all n∈ Æ .Moreover,|gn|⩽g1∈ R1(B,I),thatis,{gn}is dominated by an element belonging to R1(B,I). Thus, for any h∈+B,wehavethatI−(|gn−g0|∧h)=I∗(|gn−g0|∧h)⩽ I∗(|gn−g0|)=I∗(gn−g0)⩽I∗(gn)−I∗(g0)=I(gn)−I∗(g0)→0,whichsaysthat {gn}→g0(I−)and hence, applying the Dominated Convergence Theorem for the class R1(B,I), we deduce that g0∈R1(B,I),thatis,t0∈T(f). Let us now see some first nice consequences from Theorem 4.3. To this end we define μ∗and μ∗to be, respectively, the outer and the inner measures of μ, i.e., μ∗(A)=inf{μ(C): A⊆C, C ∈Ω} μ∗(A)=sup{μ(D): D⊆A, D ∈Ω} Proposition 5.4. If (X,B,I)is a stonian Loomis system such that I(f)= +∞ 0 I(χ[f⩾r])dr, ∀f∈+B, then Bis C0and C∞. Proof. 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