2 La dignidad humana
2.4 La dignidad humana como principio irrenunciable y absoluto
Combining equations (9.4) and (9.5) with Lemma 2.2, we have ∞ ^ n=1 ∞ _ k=n Z o X fndµ≤ Z o X f dµ≤ ∞ _ n=1 ∞ ^ k=n Z o X fndµ≤ ∞ ^ n=1 ∞ _ k=n Z o X fndµ,
which completes the proof of part (4).
§10.
Positive
E-valued measures arising from
densities
As in the real case, given a measureµandf ∈ M(X,Ω;R+), one can canonically construct a new measure from these data. In this section, we investigate the preservation of regularity of measures on locally compact Hausdorff spaces under the process. This will be an important ingredient when showing that finite measures representing positive algebra homomorphism from Cc(X)into partially ordered algebras are actually spectral measures; the latter notion
will be introduced in Definition 11.3.
The first part of this section will still be in the context of an arbitrary measure space.
Proposition 10.1. Let (X,Ω,µ,E) be a measure space, where Ω is a σ-algebra. Let f ∈ M(X,Ω;R+). For A∈Ω, set µf(A) := Z o X χAf dµ.
Thenµf :Ω→E+is a positive E-valued measure, which is finite if and only if f ∈ L1(X,Ω,µ;R). If A⊆X is aµ-null set, then A is aµf-null set.
If g∈ M(X,Ω;R+), then (10.1) Z o X gdµf = Z o X g f dµ in E.
PROOF. It is clear thatµf is a positiveE-valued measure, since the monotone convergence theorem (Theorem 9.9) shows that it isσ-additive. It is likewise clear that it is a finite measure if and only if f ∈ L1(X,Ω,µ;R), and thatµ-null sets areµf
-null sets.
For equation (10.1), considerϕ∈ E(X,Ω;R+). Choose a representationϕ=Pni=1riχAi withr1, . . . ,rn∈R+andA1, . . . ,An∈Ω. Then
Z o X ϕdµf = n X i=1 riµf(Ai) = n X i=1 ri Z o X χAifdµ = Z o X n X i=1 riχAi fdµ = Z o X ϕf dµ.
This establishes equation (10.1) for g ∈ E(X,Ω;R+). For general g ∈ M(X,Ω;R+), we choose a sequence{ϕn}∞n=1inE(X,Ω;R+)such that ϕn(x)↑ g(x)for every x in X. Then
Ro Xϕndµf ↑ Ro Xgdµ f inEby definition, whereasRo Xϕnfdµ↑ Ro Xg fdµby Theorem 9.9. Hence equation (10.1) holds forg. We shall use the following two results in the context of locally compact Hausdorff spaces. The densityf is now supposed to be bounded.
Lemma 10.2. Let(X,Ω,µ,E)be a measure space, whereΩis aσ-algebra. Let f ∈ B(X,Ω;R+)
and let{Si:i∈I}be a subset ofΩsuch that
V {µ(Si):i∈I}=0in E. Then V {µf(S i):i∈I}= 0in E.
PROOF. It is clear thatµf(S
i)≤ kfkµ(Si)inEfor alli∈I. The statement then follows from part (7) of Lemma 2.3.
Lemma 10.3. Let(X,Ω,µ,E)be a measure space, whereΩis aσ-algebra and E is a monotone complete partially ordered vector space. Letµ∈M(X,Ω,E+), and let f ∈ B(X,Ω;R+). Suppose that A∈Ωis such thatµ(A)is finite.
(1) If{Ai:i∈I}is a subset ofΩthat is upward directed and such that Ai⊆A for all i∈I
and µ(A) =_{µ(Ai):i∈I} in E, then µf(A) =_ {µf(A i):i∈I}. in E.
(2) If{Ai:i∈I}is a subset ofΩthat is downward directed and such that A⊆Aifor all
i∈I and µ(A) =^{µ(Ai):i∈I} in E, then µf(A) =^ {µf(A i):i∈I} in E.
PROOF. We prove part (1). We have
µ(A) =_{µ(Ai):i∈I}.
in E. Sinceµ(A)and then also all µ(Ai)are finite, part (5) of Lemma 2.3 and part (4) of Lemma 5.4 show that
^
{µ(A\Ai):i∈I}=0. inE. Applying Lemma 10.2, we find that
^ {µf(A
\Ai):i∈I}=0 inE. Sinceµf(A)and allµf(A
i)are finite, part (6) of Lemma 2.3 and part (4) of Lemma 5.4 show that
µf(A) =_ {µf(A
i):i∈I} inE.
We prove part (2). We have
§I.10. PositiveE-valued measures arising from densities
inE. Sinceµ(A)is finite, part (5) of Lemma 2.3 and part (4) of Lemma 5.4 show that
^
{µ(Ai\A):i∈I}=0 inE. Applying Lemma 10.2, we find that
^ {µf(A
i\A):i∈I}=0
inE. Sinceµf(A)is finite, part (6) of Lemma 2.3 and part (4) of Lemma 5.4 show that µf(A) =^
{µf(Ai):i∈I}
inE.
We shall now apply the transition fromµtoµf in the context of a locally compact Hausdorff spaceX with Borelσ-algebraB, a monotone complete partially ordered vector spaceE, and a positive E-valued measure µ:B → E+. For E =R, if µ:B →R+ is a regular Borel
measure and iff :X →R+is integrable, thenµf :B →R+is again a regular Borel measure; see[14, p. 220, Exercise 8]. The rest of this section is concerned with the vector-valued analogue of this phenomenon. These results are a key ingredient later on when establishing that a (sufficiently regular) finite measure representing a positive algebra homomorphism from Cc(X)into partially ordered algebras is actually a spectral measure; see Theorem 15.1.
The following result for finite measures is obvious from Lemma 10.3.
Corollary 10.4. Let X be a locally compact Hausdorff space, let E be a monotone complete partially ordered vector space, letµ∈M(X,B,E+), and let f∈B(X,B;R+).
(1) Ifµis a finite regular Borel measure on X , then so isµf.
(2) Ifµis a finite quasi-regular Borel measure on X , then so isµf.
We turn to the case whereµneed not be finite.
Lemma 10.5. Let X be a locally compact Hausdorff space, let E be a monotone complete partially ordered vector space, letµ ∈M(X,B,E+), let f ∈ B(X,B;R+), and let A∈ B. Set S :=
{x∈X :f(x)>0}.
(1) Suppose that at least one of the following is satisfied:
(a) µ(A)is finite andµis inner regular at A;
(b) µ(S)is finite andµis inner regular at A∩S. Thenµf is inner regular at A.
(2) Ifµ(A)is finite andµis outer regular at A, thenµf is outer regular at A.
(3) Suppose that at least one of the following is satisfied:
(a) µ(A)is finite andµis weakly inner regular at A;
(b) µ(S)is finite andµis weakly inner regular at A∩S. Thenµf is weakly inner regular at A.
PROOF. We prove part (1). The case whereµ(A)is finite is immediate from part (1) of Lemma 10.3. In the second case, we note that obviouslyµf(A) =µf(A
∩S). SinceA∩SêS has finiteµ-measure, we see, using what we have just established forA∩S, that
µf(A) =µf(A ∩S) =_ {µf(K):Kis compact andK ⊆A∩S} ≤_{µf(K):Kis compact andK⊆A}
≤µf(A). Henceµf is inner regular atA.
Part (2) is immediate from part (2) of Lemma 10.3.
We prove part (3). As with part (1), the case whereµf(A)is finite is immediate from part (1) of Lemma 10.3. In the second case, we note again thatµf(A) =µf(A
∩S), and that
A∩SêShas finiteµ-measure. Hence µf(A) =µf(A ∩S) =_ {µf((A∩S)∩U):U∈Υ} ≤_{µf(A∩U):U∈Υ} ≤µf(A) . Henceµf is weakly inner regular atA.
Our next result, again for possibly infinite measures, involves the restriction of measures to open subsets. IfUis an open subset ofX and if f ∈ M(X,B;R+), then it is easy to see that, as in the real case,
(10.2) Z o U f|UdµU= Z o X χUf dµ.
Indeed, this holds for characteristic functions, then for elementary functions, and then for f by the definition of the integral. Replacing f withχAf forA∈ BU, this yields that(µU)f|U(A) = µf(A), so that(µ
U)f|U= (µf)U. We shall writeµ f
Ufor this common resulting measure onBU.
Theorem 10.6. Suppose that E is a normal and monotone complete partially ordered vector space. Letµ∈M(X,B,E+)be such that:
(1) µis a quasi-regular Borel measure on X ;
(2) µUis a regular Borel measure on U for all U∈Υ.
Let f ∈ B(X,B;R+)be such that the subset{x∈X:f(x)>0}of X has finiteµ-measure. Then:
(1) µf is a quasi-regular Borel measure on X that is inner regular at all Borel sets; (2) µUf is a finite regular Borel measure on U for all U∈Υ.
It follows from Corollary 7.8 thatµitself is also inner regular at all Borel sets, and from Lemma 7.4 that theµUare also finite measures forU∈Υ. These are the properties that are inherited byµf.
PROOF. We start by proving thatµf is a quasi-regular Borel measure.
IfK is compact, thenK is contained in a relatively compact open subsetUofX. Since µUis a finite measure,µ(K)is finite. Since f is bounded, it is then clear thatµf(K)is finite. Henceµf is a Borel measure.
Sinceµis weakly inner regular at all Borel sets, we know from Corollary 7.8 thatµis inner regular at all Borel sets. In particular, ifA∈ B, thenµis inner regular atA∩{x∈X :f(x)>0}. Lemma 10.5 then shows thatµf is inner regular atA. Henceµf is inner regular at all Borel sets; in particular,µf is inner regular at all open subsets ofX.