3.8 Medidas de mejora para las estructuras del acueducto
4.5.1 Fuentes de abastecimiento
This chapter is based on:
A. van Rooij and W. van Zuijlen. Integrals for functions with values in partially ordered vector spaces. To appear in Positivity, preprint available athttp://arxiv. org/abs/1505.05997
Abstract
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.
Chapter
5
§5.1 Introduction
For functions with values in a Banach space there exist several notions of integration. The best known are the Bochner and Pettis integrals (see [9] and [74]). These have been thoroughly studied, yielding a substantial theory (see Chapter III in the book by E. Hille and R.S. Phillips, [48]).
As far as we know, there is no notion of integration for functions with values in a partially ordered vector space; not necessarily aσ-Dedekind complete Riesz space. In this paper we present such a notion. The basic idea is the following. (Here,E is a partially ordered vector space in which our integrals take their values.)
In the style of Daniell [23] and Bourbaki [12, Chapter 3,4], we do not start from a measure space but from a setX, a collectionΓof functionsX →E, and a functional
ϕ: Γ→E, our “elementary integral”. We describe two procedures for extendingϕto a larger class of functionsX →E. The first (see Section 5.3), the “vertical extension”, is analogous to the usual construction of the Riemann integral, proceeding from the space of simple functions. The second (see Section 5.4), the “lateral extension”, is related to the improper Riemann integral.
In Section 5.5 we investigate what happens if one repeatedly applies those exten- sion procedures, without considering the spaceE to beσ-Dedekind complete or even Archimedean. However, under some mild conditions on E one can embed E into a
σ-Dedekind complete space. In Section 5.6 we discuss the extensions procedures in the larger space. Section 5.7 and Section 5.8 treat the situation in whichΓconsists of the simpleE-valued functions on a measure space. (In Section 5.7 we haveE =R.) In Section 5.9 we consider connections of our extensions with the Bochner and the Pettis integrals for the case where E is a Banach lattice. In Section 5.10 we apply our extensions to the Bochner integral. For more alternative approaches we refer to the discussion in Section 5.11.
§5.2 Some Notation
Nis{1,2,3, . . .}.
LetX be a set. We writeP(X)for the set of subsets ofX. For a subsetAofX:
1A(x) =
(
1 ifx∈A, 0 ifx /∈A.
As a shorthand notation we write1=1X.
Let E be a vector space. We writex = (x1, x2, . . .) for functions x: N → E (i.e., elements ofEN) and we define
c00[E] ={x∈EN: ∃N ∀n≥N[xn= 0]}, c00=c00[R]
We writec0for the set of sequences inRthat converge to0,cfor the set of convergent sequences inR,`∞(X)for the set of bounded functionsX →R,`∞ for`∞(N), and
`1 for the set of absolutely summable sequences in
§5.3. The vertical extension
Chapter
5
1{n}ofRN.
For a completeσ-finite measure space(X,A, µ)we writeL1(µ)for the space of integ- rable functions,L1(µ) =
L1(µ)/
N where N denotes the space of functions that are zeroµ-a.e. Moreover we writeL∞(µ)for the space of equivalence classes of measurable functions that are bounded almost everywhere.
For a subset Γof a partially ordered vector spaceΩ, we writeΓ+=
{f ∈Γ :f ≥ 0}. If Λ,Υ⊂Ωand f ≤g for all f ∈Λ andg ∈Υwe writeΛ≤Υ; if Λ ={f} we write f ≤Υinstead of {f} ≤Υ etc. For a sequence(hn)n∈N in a partially ordered vector space we writehn↓0ifh1≥h2≥h3≥ · · · andinfn∈Nhn = 0.
§5.3 The vertical extension
Throughout this section, Eand Ωare partially ordered vector spaces,Γ⊂Ω is a linear subspace and ϕ: Γ→ E is order preserving and linear. Addi- tional assumptions are given in 5.3.14.
5.3.1 Definition. Define Γv= n f ∈Ω : sup σ∈Γ:σ≤f ϕ(σ) = inf τ∈Γ:τ≥fϕ(τ) o , (5.1) andϕv: Γv→E by ϕv(f) = sup σ∈Γ:σ≤f ϕ(σ) (f ∈Γv). (5.2)
Note: Iff ∈Ωand there exist subsetsΛ,Υ⊂ΓwithΛ≤f ≤Υsuch thatsupϕ(Λ) = infϕ(Υ), thenf ∈Γv andϕv(f) = infϕ(Υ).
5.3.2. The following observations are elementary. (a) Γ⊂Γv andϕv(τ) =ϕ(τ)for allτ ∈Γ.
(b) Γvis a partially ordered vector space andϕvis a linear order preserving map1. (c) (Γv)v= Γv and(ϕv)v=ϕv.
(d) IfΠ is a subset ofΓ, thenΠv⊂Γv.
Of more importance to us than Γv andϕv is the following variation in which we consider only countable subsets ofΓ.
5.3.3 Definition. LetΓV be the set consisting of thosef inΩfor which there exist countable setsΛ,Υ⊂Γ withΛ≤f ≤Υsuch that
supϕ(Λ) = infϕ(Υ). (5.3)
From the remark following Definition 5.3.1 it follows that ΓV is a subset of Γv and that (forf and Λ as above)ϕv(f)is equal tosupϕ(Λ). We will write ϕV =ϕv|ΓV.
We callΓV thevertical extension2 underϕofΓandϕV thevertical extension ofϕ.
1This follows from the following fact: LetA, B⊂E. IfAandB have suprema (infima) inE, then so doesA+Bandsup(A+B) = supA+ supB (inf(A+B) = infA+ infB).
2One could also define the vertical extension in caseE,Ω,Γ⊂Ωare partially ordered sets (not necessarily vector spaces) andϕ: Γ→Eis an order preserving map.
Chapter
5
In what follows we will only considerϕV and notϕv. However, most of the theory presented can be developed similarly forϕv. (For comments see 5.11.2.)
5.3.4 Example. ΓV is the set of Riemann integrable functions on [0,1] and ϕV is the Riemann integral in case E = R, Ω = R[0,1] and Γ is the linear span of
{1I :I is an interval in[0,1]}andϕis the Riemann integral onΓ.
5.3.5. In analogy with 5.3.2 we have the following. (a) Γ⊂ΓV andϕV(τ) =ϕ(τ)for allτ∈Γ.
(b) ΓV is a partially ordered vector space andϕV is a linear order preserving map. (c) (ΓV)V = ΓV and(ϕV)V =ϕV.
(d) IfΠ⊂Γ, thenΠV ⊂ΓV.
5.3.6 Definition. LetDbe a linear subspace ofE. Dis calledmediated inEif the following is true:
IfAandB are countable subsets ofD such thatinfA−B = 0inE, then
Ahas an infimum (and consequentlyB has a supremum andinfA= supB). (5.4)
D is mediated in E if and only if the following requirement (equivalent with order completeness in the sense of [72], forD=E) is satisfied
IfAandB are countable subsets of Dsuch that infA−B= 0in E, then
there exists anh∈E withB≤h≤A. (5.5)
We say thatE ismediated ifE is mediated in itself.
Note: if D is mediated in E, then so is every linear subspace of D. Every σ- Dedekind completeE is mediated, but so isR2, ordered lexicographically. Also, c00 andc0are mediated inc, butc is not mediated.
With this the following lemma is a tautology.
5.3.7 Lemma. Suppose ϕ(Γ) is mediated in E. Let f ∈ Ω. Then f ∈ ΓV if and only if there exist countable setsΛ,Υ⊂Γwith Λ≤f ≤Υsuch that
inf
τ∈Υ,σ∈Λϕ(τ−σ) = 0. (5.6)
The next example shows that ΓV is not necessarily a Riesz space even ifE andΓ are. However, see Corollary 5.3.10.
5.3.8 Example. Consider E=c,Γ =c×c,Ω =`∞×`∞. Letϕ: Γ→cbe given by ϕ(f, g) =f +g. For allf ∈ `∞ there areh
1, h2,· · · ∈ c with hn ↓ f. It follows that,ΓV ={(f, g)∈`∞×`∞:f+g∈c}. Note thatΓV is not a Riesz space since for everyf ∈`∞withf ≥0andf /∈cwe have(f,−f)∈Γ
V but(f,−f)+= (f,0)∈/ ΓV.
5.3.9 Lemma. Suppose ϕ(Γ) is mediated in E. Let Θ : Ω → Ω be an order pre- serving map with the properties:
§5.3. The vertical extension
Chapter
5
• Θ(Γ)⊂ΓV. ThenΘ(ΓV)⊂ΓV.
Proof. Let f ∈ΓV and let Λ,Υ ⊂Γ be countable sets with Λ ≤f ≤ Υsatisfying (5.6). ThenΘ(Λ)≤Θ(f)≤Θ(Υ)and inf τ∈Θ(Υ),σ∈Θ(Λ)ϕ(τ−σ) =τ∈Υ,σinf∈Λϕ Θ(τ)−Θ(σ) ≤ inf τ∈Υ,σ∈Λϕ(τ−σ) = 0. (5.7)
5.3.10 Corollary. Suppose thatϕ(Γ)is mediated inE. SupposeΩis a Riesz space andΓ is a Riesz subspace ofΩ. Then so isΓV.
Proof. Apply Theorem 5.3.9 withΘ(ω) =ω+.
5.3.11. IfΓ is a directed set, i.e., Γ = Γ+
−Γ+, then so isΓ
V. Indeed, iff ∈ΓV, then there existσ, τ ∈Γ+ such thatf
≥τ−σand thusf = (f+σ)−σ∈Γ+V −Γ+V.
5.3.12. In the last part of this section we will consider a situation in which Ωhas some extra structure. But first we briefly consider the case where E is a Banach lattice with σ-order continuous norm. As it turns out, such an E is mediated (see Theorem 5.4.24), but is not necessarily σ-Dedekind complete (consider the Banach lattice C(X) where X is the one-point compactification of an uncountable discrete space). For suchE we describeΓV in terms of the norm.
5.3.13 Theorem. Let E be a Banach lattice with a σ-order continuous norm. Let
Ωbe a Riesz space andΓbe a Riesz subspace ofΩ. Forf ∈Ωwe have: f ∈ΓV if and only if for everyε >0 there existσ, τ ∈Γwith σ≤f ≤τ andkϕ(τ)−ϕ(σ)k< ε. Proof. First, assume f ∈ ΓV. As Γ is a Riesz subspace of Ω there exist sequences
(σn)n∈Nand(τn)n∈Nin Γsuch thatσn ↑,τn↓,
σn ≤f ≤τn (n∈N), sup n∈N
ϕ(σn) = inf
n∈Nϕ(τn). (5.8) Thenϕ(τn−σn)↓0in E, sokϕ(τn)−ϕ(σn)k ↓0 and we are done.
The converse: For eachn∈N, chooseσn, τn∈Γ for which
σn ≤f ≤τn, kϕ(τn)−ϕ(σn)k ≤n−1. (5.9) Setting σ0
n=σ1∨ · · · ∨σn andτn0 =τ1∧ · · · ∧τn we have, for eachn∈N
σ0n, τn0 ∈Γ, σ0n≤f ≤τn0. (5.10) If n ≥ N, then 0 ≤ σ0
n−σ0N ≤ f −σN ≤ τN −σN, whence kϕ(σ0n)−ϕ(σN0 )k ≤
kϕ(τN)−ϕ(σN)k ≤ N−1. Thus, the sequence (ϕ(σ0n))n∈N converges in the sense of the norm. So does(ϕ(τ0
n))n∈N. Their limits are the same elementaofE, and, since
Chapter
5
5.3.14. In the rest of this section Ω is the collection FX of all maps of a
set X into a partially ordered vector space F.
5.3.15. A function g :X → Rdetermines a multiplication operator f 7→gf in Ω. We investigate the collection of all functionsg for which
f ∈ΓV =⇒ gf∈ΓV, (5.11)
and, for givenf, the behaviour of the mapg7→ϕV(gf).
5.3.16. For an algebra of subsets ofX, A ⊂ P(X)we write [A]for the Riesz space of all A-step functions, i.e., functions of the form Pni=1λi1Ai for n ∈ N, λi ∈ R,
Ai∈ Afori∈ {1, . . . , n}. Define the collection of functions[A]o by
[A]o={f ∈RX: there are (sn)n∈Nin [A]and(jn)n∈Nin [A]+ (5.12) for which|f−sn| ≤jn andjn↓0pointwise}.
(This [A]o is the vertical extension of [
A] obtained by, in Definition 5.3.3, choosing
E = RX,Ω = RX,Γ = [A], ϕ(f) = f (f ∈ Γ).) Note that [A] and [A]o are Riesz spaces, and uniform limits of elements of [A] are in [A]o. (Actually, [A]o is uniformly complete.) Furthermore, [A]o contains every bounded function f with
{x∈X :f(x)≤s} ∈ Afor alls∈R. In case Ais aσ-algebra, [A]o is precisely the collection of all boundedA-measurable functions.
5.3.17 Lemma. Let A ⊂ P(X) be an algebra of subsets of a set X. Suppose that
(gn)n∈Nis a sequence in[A]ofor whichgn ↓0pointwise. Then there exists a sequence
(jn)n∈N in[A] withjn≥gn andjn ↓0 pointwise.
Proof. For all n ∈ N there exists a sequence (snk)k∈N in [A] with snk ≥ gn for all
k ∈ N and snk ↓k gn pointwise. Since (gn)n∈N is a decreasing sequence, we have
smk≥gn for allm≤nand allk∈N. Hencejn:= infm,k≤nsmk is an element in[A] withjn≥gn. Clearlyjn ↓andinfn∈Njn= infn∈Ninfm,k≤nsmk= infn∈Ninfk∈Nsnk=
infn∈Ngn= 0.
The following lemma is a consequence of Lemma 5.3.9.
5.3.18 Lemma. Define the algebra
A={A⊂X :f1A∈Γ forf ∈Γ}. (5.13) If ϕ(Γ)is mediated inE, then
f1A∈ΓV (f ∈ΓV, A∈ A). (5.14)
5.3.19 Definition. E is called Archimedean3 (see Peressini [73]) if for alla, b∈E
the following holds: ifna≤b for alln∈N, thena≤0.
§5.3. The vertical extension
Chapter
5
5.3.20 Definition. A sequence (an)n∈N in E is called order convergent to an ele- menta∈E if there exists a sequence(hn)n∈NinE+withhn↓0and−hn ≤a−an≤
hn.
Notation: an −→o a.
5.3.21 Theorem. Let Abe as in (5.13). Suppose thatE is Archimedean,Γ is dir- ected and ϕ(Γ)is mediated inE. Furthermore assume ϕhas the following continuity property.
IfA1, A2, . . . inAare such thatA1⊃A2⊃ · · · andTn∈NAn=∅, (5.15) thenϕ(f1An)↓0 for allf ∈Γ
+.
(a) gf ∈ΓV for allg∈[A]o and allf ∈ΓV. (b) Letg∈[A]o and let(g
n)n∈Nbe a sequence in[A]ofor which there is a sequence
(jn)n∈N in[A]o+ with −jn≤gn−g≤jn andjn↓0 pointwise. Then
ϕV(gnf) o
−
→ϕV(gf) (f ∈ΓV). (5.16)
(Order convergence in the sense of E.) Proof. We first prove the following:
(?) Letf ∈Γ+V. Let (gn)n∈Nbe a sequence in[A]o for whichgnf ∈ΓV for alln∈N andgn ↓0 pointwise. Then
ϕV(gnf)↓0. (5.17)
Letσ∈Γ+,σ
≥f. It follows from Lemma 5.3.17 that we may assumegn ∈[A]for all
n∈N. For alln∈Nwe have0≤ϕV(gnf)≤ϕV(gnσ), so we are done ifϕV(gnσ)↓0. Leth∈E,h≤ϕV(gnσ)for alln∈N; we proveh≤0.
Takeε >0. For eachn∈N, setAn={x∈X :gn(x)≥ε}. ThenAn ∈ Aforn∈N andA1⊃A2⊃ · · · andTn∈NAn=∅. PuttingM =kg1k∞we see that
gn ≤ε1X+M1An (n∈N), (5.18)
whence
h≤ϕV(gnσ)≤εϕ(σ) +M ϕ(1Anσ) (n∈N). (5.19)
By the continuity property of ϕ,h≤εϕ(σ). As this is true for each ε >0 andE is Archimedean, we obtainh≤0.
(a) SinceΓV is directed (see 5.3.11) it is sufficient to considerf ∈ΓV+. Letg∈[A]o. There are sequences of step functions(hn)n∈Nand(jn)n∈Nfor whichhn↑g,jn↓gand thusjn−hn↓0. By Lemma 5.3.18hnf, jnf ∈ΓV for alln∈N. Thenhnf ≤gf ≤jnf for n∈ Nand infn∈NϕV((jn−hn)f) = 0 by (?). By Lemma 5.3.7 and 5.3.5(c) we obtain thatgf ∈ΓV.
(b) It is sufficient to consider f ∈ Γ+V. By (a) we may also assumeg = 0. But then (b) follows from (?).
Chapter
5
5.3.22 Remark. Consider the situation in Theorem 5.3.21. SupposeB ⊂ Ais aσ- algebra. Then all boundedB-measurable functions lie in[A]o. If(g
n)n∈Nis a bounded sequence of bounded B-measurable functions that converges pointwise to a function
g, then the condition of Theorem 5.3.21(b) is satisfied.
5.3.23 Remark. In the next section we will consider a situation similar to the one of Theorem 5.3.21, in which Ais replaced by a subsetI that is closed under taking finite intersections. We will also adapt the continuity property on ϕ(see 5.4.3).
§5.4 The lateral extension
The construction described in Definition 5.3.3 is reminiscent of the Riemann integral and, indeed, the Riemann integral is a special case (see Example 5.3.4).
In the present section we consider a type of extension, analogous to theimproper Riemann integral. One usually defines the improper integral of a functionf on[0,∞)
to be lim s→∞ Z s 0 f(x) dx, (5.20)
approximating the domain, not the values off.
For our purposes a more convenient description of the same integral would be ∞ X n=1 Z an+1 an f(x) dx, (5.21)
where0 =a1< a2<· · · andan → ∞. Here the domain is split up into manageable pieces.
Splitting up the domain is the basic idea we develop in this section. (This may explain our use of the terms “vertical” and “lateral”.)
Throughout this section, X is a set, E and F are partially ordered
vector spaces, Γ is a directed4 linear subspace of FX, and ϕ is a linear
order preserving map Γ→E. (With Ω =FX, all considerations of Section
5.3 are applicable.)
Γ⊂FX
E
ϕ
Furthermore, I is a collection of subsets of X, closed under taking
finite intersections. See Definition 5.4.1 and Definition 5.4.2 for two more assumptions.
4For the construction of the lateral extension, one does not need to assume thatΓis directed. However, as one can see later on in the construction, the only part ofΓthat matters for the extension isΓ+−Γ+.
§5.4. The lateral extension
Chapter
5
As a shorthand notation, if (an)n∈Nis a sequence inE+ and{PNn=1an:N ∈N} has a supremum, we denote this supremum by
X
n
an. (5.22)
5.4.1 Definition. A disjoint sequence(An)n∈N of elements in I whose union is X is called apartition. If (An)n∈Nand (Bn)n∈N are partitions and for alln∈N there exists anm∈Nfor whichBn⊂Am, then(Bn)n∈Nis called arefinement of(An)n∈N. Note that if(An)n∈Nand(Bn)n∈Nare partitions then there exists a refinement of both
(An)n∈N and(Bn)n∈N (e.g., a partition that consists of all sets of the formAn∩Bm withn, m∈N).
We assume that there exists at least one partition.
5.4.2 Definition. We call a linear subspace∆ ofFX stable (under
I) if
f1A∈∆ (f ∈∆, A∈ I). (5.23)
If∆ is a stable space, then a linear and order preserving mapω : ∆→E is said to belaterally extendable if for all partitions(An)n∈N
ω(f) =X
n
ω(f1An) (see(5.22)) (f ∈∆
+). (5.24)
We assume Γ is stable andϕ is laterally extendable.
5.4.3. In the situation of Theorem 5.3.21 we can choose I = A; then (5.15) is precisely the lateral extendability of ϕ.