Absolute continuity theorems for abstract Riemann integration

ABSOLUTE CONTINUITY THEOREMS FOR ABSTRACT RIEMANN INTEGRATION E. de Amo, Almería, R. del Campo, Almería, M. Díaz-Carrillo, Granada Abstract. Absolute continuity for functionals is studied in the context of proper and abstract Riemann integration examining the relation to absolute continuity for finitely additive measures and giving results in both directions: integrals coming from measures and measures induced by integrals. To this end, we look for relations between the corresponding integrable functions of absolutely continuous integrals and we deal with the possibility of preserving absolute continuity when extending the elemental integrals. Keywords: finitely additive integration, abstract Riemann integration, absolute continuity 1. Introduction It is well known that there are two classical ways of developing an Integration Theory: On the one hand, there is the set theoretic starting point, which we will denote as (µ/Ω): Xis a non empty set, Ωis a σ-algebra of the power set of Xand µis a measure on Ω. In this context, standard and classical methods lead to the L1(Ω, µ) class of the Lebesgue integrable functions (see [11]). On the other hand, there exists a functional setting which we will denote as (I/B): The starting point here is a Daniell Loomis system, that is a triple (X, B, I)where Bis a vector lattice of real functions defined on Xand Iis a Daniell integral on B({hn} ⊆ B,hn↓0⇒I(hn)→0). In this case we get the corresponding class L1(B, I)of Daniell integrable functions. For a recent account of the functional extension procedures we refer the reader to [6]. Both contexts have a common hypothesis which plays the central role: continuity. For the (µ/Ω) context it is the σ-additivity of µand for the (I/B)setting it is the Daniell (or Bourbaki) condition on I. The interplay between these two schemes, (µ/Ω) and (I/B), is well known: We obtain the corresponding Loomis system (X, BΩ, Iµ)induced by the measure space (X, Ω, µ)and, when Bis stonean (i.e., 1∧B⊆B) the Loomis system (X, B, I) induces the corresponding measure space (X, ΩB, µI). A classical text which clearly shows these facts is the book by Pfeffer [15]. When the continuity of the measure is dropped (and we work without or with weaker continuity conditions) two new paradigms arise: the class R1(µ)of the abstract Riemann µ-integrable functions with respect to a finitely additive measure µ (see [13]) versus its functional analogue, the class R1(B, I)of the abstract Riemann I-integrable functions in [9]. We can trace this back to the works of Loomis [14] and Aumann [4] on integral extension of positive linear functionals. For those attempts there are no Lebesgue convergence theorems. In this functional context, the class R1(B, I)of the abstract Riemann I-integrable functions was obtained. For this class it is possible to establish results such as Lebesgue convergence type theorems and the usual characterizations of integrability. Moreover, a unified treatment for the Dunford-Schwartz, abstract µ-Riemann, Daniell and Bourbaki integrals is achieved (see [9] and [10]). The papers [7]–[10] by Díaz-Carrillo and Günzler, and by Díaz-Carrillo and Mu˜nozRivas are the references for this approach. This will be the framework for what follows. 2. Preliminaries For R := R ∪{−∞,+∞}, where R is the real line, we extend the usual addition in R to R 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 R Xconsist of all functions defined on X with values in R . All operations and relations in R Xare defined pointwise, with the convention inf ∅:= +∞and sup ∅:= −∞. A functional T: R X−→ R will be called subadditive if T(f+g)⩽T(f) + T(g) for all f, g ∈ R Xbut T(f) = −T(g) = +∞and T(f) = −T(g) = −∞. The notion of a superadditive functional is introduced in a completely dual way. A triple (X, B, I)is called a Loomis system if B⊆ R Xis a vector lattice of R is a positive linear functional. We set +B:= {h∈ real functions and I : B → B : h ⩾ 0}. Given (X, Ω, µ)with µa finite finitely additive measure and Ωa ring, we call (X, BΩ, Iµ)the induced Loomis system, where BΩis the vector lattice of µ-simple functions, BΩ:= h∈ R X:h= n X i=1 aiχAi, ai∈ R , Ai∈Ω, µ([h6= 0]) <+∞, and Iµis its canonical elemental integral given by Iµ(h) := n X i=1 aiµ(Ai),∀h∈BΩ. 3. Proper and abstract Riemann integration Let (X, B, I)be a Loomis system. For f∈ R X, following Loomis in [14] we define by I−(f) := inf{I(h): h∈B, h ⩾f}, I+(f) := sup{I(h): h∈B, h ⩽f} the corresponding upper and lower integrals of f, which verify −∞ ⩽I+(f)⩽ I−(f)⩽+∞,∀f∈ R X,I−is subadditive, I+is superadditive, and both are positively homogeneous. The class of the properly Riemann integrable functions is defined by Rprop(B, I) := {f∈ R X:I+(f) = I−(f)∈ R }, or, equivalently, by Rprop(B, I) = {f∈ R X:∀ε > 0,∃h, g ∈B, h ⩽f⩽gand I(g−h)< ε} and it is a vector lattice where the functional I:= I+=I−is linear and increasing, i.e., it is an integral which extends the original I. For this class there are no satisfactory Lebesgue convergence type theorems to make a consistent Integration Theory. Therefore, it is necessary to introduce a “local convergence” to ensure this kind of results. The local I-convergence for sequences of functions {fn}in R Xto a function fin R X, denoted by {fn}−→f(I−), means that {I−(|fn−f|∧h)} → 0,∀h∈+B, and it has been used in [9] to define the class R1(B, I)of the abstract Riemann integrable functions as R1(B, I) := {f∈ R X:∃{hn}in B,I-Cauchy;{hn}−→f(I−)} where {hn}I-Cauchy means that I(|hn−hm|)→0, for n, m →+∞. Moreover, for f∈R1(B, I)we set I(f) := lim n→+∞I(hn)for any sequence {hn}in B I-Cauchy and such 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. Further relations between the classes Rprop(B, I)and R1(B, I)are given in [9] by the following characterizations: f∈R1(B, I)⇔f±∧h∈Rprop,∀h∈+Band I+(|f|)<+∞.(1) f∈Rprop(B, I)⇔f∈R1(B, I)and ∃h∈+B:|f|⩽h.(2) In fact, in [9, Th.1.6], it is proved that (3) I(f) = I+(f),∀f∈+R1(B, I). We recall that the class of the null-functions is introduced in this context by N1(B, I) := {f∈R1(B, I): I(|f|) = 0} or, equivalently, by N1(B, I) = {f∈ R X:I−(|f|∧h) = 0,∀h∈+B}. On the other hand, the localized functional I− lin the sense of [17] 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 [10] guarantees that R1(B, I)is the closure of B in R Xwith respect to the integral seminorm I− l(|·|)and I− l(f) = I(f),∀f∈R1(B, I)(Iis the only I− l-continuous extension of Ifrom Bto R1(B, I)). Finally, we consider the functional I∗(f) := inf{I(g): g∈R1(B, I), g ⩾f}, which is also positively homogenous, monotone, subadditive and, evidently, extends Ifrom R1(B, I)to R X. Definition 3.1. A Loomis system (X, B, I)is called C+∞or upper continuous if lim r→+∞I∗(f−f∧r) = 0,∀f∈+B. Upper continuity on Bis hereditary for the class R1(B, I); that is: Lemma 3.2. If (X, B, I)is C+∞, then so is R1(B, I). In general R1(B, I)need not be closed under multiplication, but we will use the following two facts which can be easily checked. Lemma 3.3. If BB ⊆B,f∈R1(B, I)and k∈Bis bounded, then fk ∈ R1(B, I). Lemma 3.4. If (X, B, I)is a C+∞Loomis system and hand χAare in R1(B, I) then so is hχA. There are three basic theorems to obtain a good Measure and Integration Theory: Lebesgue, Fubini and Radon-Nikodym type theorems. For the class R1(B, I), Lebesgue theorems were given by Díaz-Carrillo and Mu˜noz-Rivas in [9] and Fubini type theorems were found by de Amo and Díaz-Carrillo in [3]. Partial attempts in order to obtain Radon-Nikodym type theorems were done by de Amo, Chit¸escu and Díaz-Carrillo (see [1] and [2]). We will now study the notion of absolute continuity in this functional setting of proper and abstract integration and its relations to the notion of absolute continuity for finitely additive measures. 4. Absolute continuity We recall that, given two finitely additive measures µand νon a ring Ω,νis said to be absolutely continuous with respect to µ, and is denoted by ν≪µ, if ∀ε > 0,∃δ > 0: A∈Ω, µ(A)< δ ⇒ν(A)< ε (see Bochner [5, p. 778], Fefferman [12, p. 35], Dunford-Schwartz [11, p. 131]). This definition clearly implies the classical one, µ(A) = 0 ⇒ν(A) = 0,∀A∈Ω, and both are, in fact, equivalent when µand νare measures such that ν(A)<+∞ for all A∈Ωwith µ(A)<+∞. The most natural transcription for absolute continuity to the analogous functional context (I/B)is the following one: Let Iand Jbe two positive functionals on B. We say that Jis I-continuous (continuous with respect to I) if ∀ε > 0,∃δ > 0: h∈+B, I(h)< δ ⇒J(h)< ε. Unfortunately, this definition fails since I-continuity is, in fact, a kind of boundedness condition. Proposition 4.1. Let (X, B, I)be a Loomis system and Ja positive functional on B. The following conditions are equivalent: (i) Jis I-continuous; (ii) ∃M > 0: J(h)⩽MI(h),∀h∈+B(J⩽MI, for abbreviation). This equivalence allows us to show that integrals induced by absolute measures need not be continuous in this sense, that is, there exist measures µand νsuch that ν≪µbut Iνis not Iµ-continuous. Example 4.2. Let X= [0,1], let λbe the Lebesgue measure in Xand νthe measure given by ν(A) = RAhdλ, where h:X−→ R is the λ-integrable function defined by h(0) = 0 and h(t) = 1/√t,∀t6= 0. Evidently, µis absolutely continuous with respect to λ, but there is no positive Msuch that Jν⩽MIλ: If we assume that such an Mexists then, in particular, we have ν([0,1/n]) ⩽Mλ([0,1/n]),∀n∈ N , that is Z1/n 0 h(t) dt⩽M1 n⇒n⩽M2 4,∀n∈ N , which leads to contradiction. Therefore we have to weaken I-continuity in order to define a satisfactory notion of absolute continuity for functionals. The latter was introduced in [1] and reads as follows: Definition 4.3. Let (X, B, I) be a Loomis system and J a positive functional on B. J is said to be absolutely I-continuous (absolutely continuous with respect to I) and is denoted by J ≪ I, if ∀ε > 0, ∀h ∈ +B, ∃δ > 0 : ∀k ∈ +B, k ⩽ h, I(k) < δ ⇒ J(k) < ε. The next theorem makes evident when absolutely continuous finitely additive measures yield absolutely continuous elementary integrals. Theorem 4.4. Let µand νbe finitely additive measures such that ν(A)<+∞ for all A∈Ωwith µ(A)<+∞. If ν≪µthen Iν≪Iµ. P r o o f. Assume that ν≪µ, let ε > 0and f∈+BΩ. There are ai>0 and pairwise disjoint Ai∈Ωsuch that f= n P i=1 aiχAi. Set A:= n S i=1 Ai∈Ωand β:= sup{ai:i= 1, . . . , n}>0. Note that µ(A)<+∞, since µ([f6= 0]) <+∞. If ν(A) = 0, then Iν(f)⩽βν(A) = 0 and therefore Iν(h)⩽Iν(f) = 0 < ε, ∀h∈ +BΩwith h⩽f. Assume that ν(A)>0and let α:= 1 2ε/ν(A)>0. Since ν≪µ, there exists  > 0 such that ∀E⊆Ω, µ(E)<  ⇒ν(E)<ε 2β. Let δ:= α > 0and h∈+BΩwith h⩽fand Iµ(h)< δ. There are ei>0and pairwise disjoint Ei∈Ωsuch that h= m P j=1 ejχEj. Moreover, since h⩽fwe have E:= m S j=1 Ej⊆ n S i=1 Ai=Aand ej⩽β, ∀j= 1, . . . , m. Let us now consider sets T:= {t∈ N : 1 ⩽t⩽m, et< α}, S:= {t∈ N : 1 ⩽t⩽m, et⩾α}, which are disjoint with S∪T={1,...,m}, and define functions h1:= X t∈T etχEtand h2:= X s∈S esχEs. Evidently h1, h2∈+BΩand h=h1+h2. Furthermore, Iν(h1)< α X t∈T ν(Et)⩽αν(A) = ε 2 and an easy computation shows that µS s∈S Es< δ/α =. Hence, νS s∈S Es< 1 2ε/β and, consequently, Iν(h2) = X s∈S esν(Es)⩽βν[ s∈S Es<ε 2. Therefore Iν(h) = Iν(h1) + Iν(h2)< ε, which completes the proof.  5. Absolute continuity and proper Riemann integration In this section we will study the good behaviour of absolute continuity with respect to proper Riemann integration. The first result says that absolute continuity of Jwith respect to Itransfers convergence to 0for B-bounded sequences from the integral Ito the integral J: Lemma 5.1. Assume that J≪Iand let {hn}be a sequence in +Bsuch that there exists h∈+Bwith hn⩽h, ∀n∈ N and I(hn)→0. Then J(hn)→0. In particular, we have the following facts: Corollary 5.2. If J≪Iand Iis Daniell, then so is J. Corollary 5.3. If J≪Iand {hn}is an I-Cauchy sequence in +Bsuch that there exists h∈+Bwith |hn−hm|⩽h∀n, m ∈ N , then {hn}is J-Cauchy. Theorem 5.4. If J≪I, then (i) Rprop(B, I)⊆Rprop(B, J), (ii) J≪Ion Rprop(B, I). P r o o f. (i) Let f∈Rprop(B, I)and ε > 0. There are kε, hε∈Bsuch that kε⩽f⩽hεand I(hε−kε)< ε. For ε > 0and hε−kε∈+B, since J≪I, there exists δ > 0such that ∀g∈+B, g ⩽hε−kε, I(g)< δ ⇒J(g)< ε. We can also find kδ, hδ∈Bsuch that kδ⩽f⩽hδand I(hδ−kδ)< δ. Since δdepends only on ε, we can consider the functions: k′ ε:= kε∨kδand h′ ε:= hε∨hδ, which verify that k′ ε, h′ ε∈Band k′ ε⩽f⩽h′ ε. Therefore, we have found that h′ ε − k′ ε ∈ +B, h′ ε − k′ ε ⩽ hε − kε with I(h′ ε − k′ ε) ⩽ I(hδ − kδ) < δ and, consequently, J(h′ ε − k′ ε) < ε. Thus f ∈ Rprop(B, J). (ii) Assume that J≪Iand let ε > 0. Given f∈+Rprop(B, I)we can take kε=kε(f)∈+Bsuch that f⩽kε.I-continuity of Jgives δ=δ(ε, f)>0such that ∀h∈+B, h ⩽kε, I(h)< δ ⇒J(h)< ε. Let := 1 2δ. Given g∈+Rprop(B, I),g⩽fwith I(g)<  there are hδ, kδ∈+B with hδ⩽g⩽kδand I(kδ−hδ)< . Taking hε:= kε∧kδ∈+B, we have g⩽kε∧kδ=hεand I(hε)⩽I(kδ)⩽I(kδ−hδ) + I(hδ)<  +I(g)<  +=δ. Therefore, we deduce that J(hε)< ε and hence J(g) = J−(g) = inf{J(h): h∈+B, g ⩽h}⩽J(hε)< ε, that is, J≪Ion Rprop(B, I). We are able to give a first sufficient condition for finitely additive measures induced by absolutely continuous integrals to be absolutely continuous. Given two positive functionals Iand J, let (X, Ω(I), µI)and (X, Ω(J), νJ)be their respective finitely additive measure induced spaces, that is, Ω(I) = {A⊆X:χA∈R1(B, I)}, µI(A) = I(χA),∀A∈Ω, Ω(J) = {A⊆X:χA∈R1(B, J)}, νJ(A) = J(χA),∀A∈Ω. Proposition 5.5. If 1∈Rprop(B, I)and J≪I, then νJ≪µI(on Ω(I)∩Ω(J)). P r o o f. Since J≪I, Theorem 5.4 says that J≪Ion Rprop(B, I)⊆ Rprop(B, J). Thus, for ε > 0and 1∈Rprop(B, I)there exists δ > 0such that ∀h∈Rprop(B, I)with h⩽1and I(h)< δ ⇒J(h)< ε. Given A∈Ω(I)∩Ω(J)with µI(A)< δ we have χA∧h∈Rprop(B, I),∀h∈+B, χA∧h⩽1and I(χA∧h)⩽I(χA) = µI(A)< δ. Therefore, it follows that J(χA∧h)< ε, ∀h∈+Band, keeping in mind that χA∈+R1(B, J), we conclude that νJ(A) = J(χA) = J− l(χA) = sup{J−(χA∧h): h∈+B}< ε.  In the following section, the condition 1∈Rprop(B, I)will be relaxed to 1∈ R1(B, I)and Ω(I)∩Ω(J)will be, in fact, Ω(I)(see Corollary 6.9).