Notes on Tensor Product Measures

0.22 MB·

by  Shigeru Yamagami

Notes on Tensor Product Measures
7
1
YAMAGAMI Shigeru
0
2 Nagoya University
Graduate School of Mathematics
n
a
J January 13, 2017

2
1

] Introduction
A
F ConsideraspectraldecompositionU(t)= ReitτE(dτ)ofaone-parameteruni-
h. tarygroupU(t)onaHilbertspaceH,whereE()isaprojection-valuedmeasure
t on R. In quantum mechanics, the dynamRical b·ehavior of a physical system is
a
m described by the associated automorphic action of R on the algebra L(H) of
bounded linear operators. The relatedtransitionprobabilities are associatedto
[
(ξ U(t)TU(t)∗η) = (U(t)∗ξ TU(t)∗η) (ξ,η H, T L(H)), which takes the
| | ∈ ∈
1 form
v ( e−itτE(dτ)ξ T e−itτ′E(dτ′)η)
5 R | R
1 Z Z
2 in terms of the spectral measure. At first glance, it seems quite natural to
3 rewrite this to the product measure form like
0
1. eit(τ−τ′)(E(dτ)ξ TE(dτ′)η),
R R |
0 Z ×
7
whichmeansthatweexpectacomplex-valuedmeasure(E(dτ)ξ TE(dτ′)η)tobe
:1 well-defined on R2. This anticipation is reasonably generalized|to the following
v question: LetT L(H)andξ(),η()beH-valuedmeasuresonaσ-algebraBin
Xi a setS. Itis imm∈ediate to chec·k tha·tthe mapB B A B (ξ(A)Tη(B))
r is extended to a finitely additive function µ on×the∋Boo×lean7→algebra| B B
a generated by B B. Is it then possible to extend µ to a complex measure⊗on
theσ-algebragen×eratedbyB B? WhenT isafiniterankoperator,µcertainly
×
admitssuchanextensionasalinearcombinationofproductmeasuresandwith
a litte more effort we can show that the question is answered affirmatively for
a trace class operator. The general answer, however, turns out to be negative:
A bounded linear operator T has the measure extension property if and only if
T is in the Hilbert-Schmidt class ([Swartz1976,Theorem 8]1 ).
Our main purpose here is to collect relevant results together and combine
them to give a self-contained proof of it.
1Wewouldliketopointout,however,thattheproofthereisbasedonatheoreminanother
paper,whichseemsdifficulttobeidentified.
1
Notation: ForaBanachspaceV,itsunitballisdenotedbyV anditsdualspace
1
by V∗. Given a set T, ℓ∞(T) denotes the Banach space of bounded complex-
valued functions on T with the sup-norm, which is the dual Banach space of
ℓ1(T) of summable functions. We then have a canonical isometric embedding
V ℓ∞(V∗) for a Banach space V as a restriction of the canonical pairing
V →V∗ C1: For v V, v ℓ∞(V∗) is defined by v(v∗)= v,v∗ (v∗ V∗).
× → ∈ ∈ 1 h i ∈ 1
More generally, if T V∗ satisfies v = sup v,v∗ ;v∗ T , then the
⊂ 1 k k {|h i| ∈ }
restrictionmapℓ∞(V1∗)→bℓ∞(T)is isometric onV b={v;v ∈V} andwe getan
embedding V ℓ∞(T).
→
The semi-variation of a finitely additive measubre φbis denoted by φ, while
| |
φ is set aside to designate the total semi-variation of φ.
k k
1 Vector Valued Measures
WeshallmainlydealwithBanachspacesasvectorspacesandnominate[Diestel-
Uhl] as a basic reference. See also [Ricker, Chap.1] for a friendly survey on the
subject.
In a (Hausdorff) topological vector space V, a family of vectors v is
i i∈I
{ }
said to be summable if we can find a vector v V fulfilling the following
∈
condition: Given any neighbourhood N of v, we can find a finite subset F I
⊂
sothat v N foranyfinite subsetF′ I F. Thevectorv isunique
j∈F∪F′ j ∈ ⊂ \
if it exists and denoted as v = v .
P i∈I i
When V is a Fr´echet space, the condition is equivalent to the following:
P
Given any neighborhood N of 0, we can find a finite subset F of I so that
v N for any finite subset F′ I F.
j∈F′ j ∈ ⊂ \
When I is countable, any counting labeling i ;n 1 gives
n
P { ≥ }
n
v = lim v .
n→∞ ik
k=1
X
Conversely, in a Banach space V, if any counting labeling satisfies the above
convergence relation, then v is summable and v = v . In fact, if not,
{ i} i∈I i
P
ǫ>0, F ⋐I, F′ ⋐I F, v ǫ
j
∃ ∀ ∃ \ ≥
(cid:13)jX∈F′ (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
andwe canfind apartition F ofI byfinite subsetssatisfying v
n k j∈Fn jk≥
ǫ. Let i be a counting labeling adapted to the increasing sequence F
k 1
{ } F P ⊂
F F . Then v cannot be a Cauchy sequence by looking at
1 ∪ 2 ⊂ ··· { ik}k≥1
k = F + + F (n=1,2, ).
1 n
| | ··· | | ···
WhenV isfinite-dimensionalwith anycompatiblenorm,thesummability
k·k
of v is equivalent to v < , the so-called absolute convergence.
{ i}i∈I i∈Ik ik ∞
In fact, for a basis v∗, ,v∗ , the summability implies absolute convergence
of v∗(v ) for 1{ 1 j···Pn,n}which is equivalent to n v∗(v ) < .
i∈I j i ≤ ≤ j=1 i∈I| j i | ∞
Note that n v∗(v) (v V) defines a norm on V.
P j=1| j | ∈ P P
P
2
Example 1.1. Let δ be an ONB in a separable Hilbert space H. Then,
n n≥1
for ξ H, (δ ξ)δ{ } is summable and ξ = (δ ξ)δ , whereas its
∈ { n| n}n≥1 n≥1 n| n
absoluteconvergenceisequivalenttothestrongercondition (δ ξ) < .
P n≥1| n| | ∞
LetV beaBanachspaceandBbeaBooleanalgebrainaPsetS. AV-valued
semi-measure is an additive map φ : B V. We say that φ is countably
∞ →
additive if A = A is a countable partition in B, then φ(A) = φ(A ).
n n
n=1 n≥1
G X
Bythe correspondancebetweenB = A =A ( A )andA =
B B , countable additivity ismequivanl≥emnt tno the c\ond1it≤ionn<:mIf Bn innB,
n n+1 n
\ F S ↓ ∅
then lim φ(B )=0.
n→∞ n
A semi-measure φ:B V is called a measure if B is a σ-algebra and φ is
→
countably additive.
Clearly countable additivity implies lim φ(A ) = 0. We say that a
n→∞ n
semi-measureφissqueezing2 iflim φ(A ) =0foranydisjointsequence
n→∞ n
A in B ( A B being not assumked). Rkemark that a squeezing semi-
n n≥1 n n
m{ eas}ure φ is co∪ntinuou∈s, i.e., A in B implies lim φ(A ) = 0, and, if B is
n n n
↓ ∅
a σ-algebra, a continuous semi-measure is squeezing. This squeezing property
together with finite additivity of φ in turn assuresthe summability of φ(A ) .
n
{ }
In fact, non-summability
n
ǫ>0, N, n m N, φ(A ) ǫ
k
∃ ∀ ∃ ≥ ≥ ≥
(cid:13)kX=m (cid:13)
(cid:13) (cid:13)
allowsustofindasubsequence1=l <l <(cid:13) satisfyin(cid:13)g φ(A )
1 2 ··· k lj≤k<lj+1 k k≥
ǫandwegetanon-squeezingseries ∞ φ(B ),whereB = A gives
a disjoint sequence in B. Notice thatjφ=(1B )=j j φP(∪Alj)≤kb<yljfi+n1itekaddi-
P j lj≤k<lj+1 k
tivity of φ.
P
Example 1.2. Let B be the power set of N. Then an additive map φ : B
→
V gives rise to a sequence v = φ( n ) and the squeezing property of φ
n
{ { } }
impliesthe Cauchyconditionthat,givenǫ>0,thereexistsanN 1satisfying
≥
v ǫ for any finite subset F of N+1,N+2, . Converselygiven
k j∈F jk≤ { ···}
a sequence v satisfying the Cauchy condition, v is summable for any
n n n∈A
suPbset A {N a}nd a countably additive map φ:B{ }V is defined by
⊂ →
φ(A)= v .
n
n∈A
X
Example 1.3. Let B be the Boolean algebra generated by finite subsets of N:
A B if and only if either A or N A is finite. A semi-measure φ : B Z is
∈ \ →
then defined by
A if A < ,
φ(A)= | | | | ∞
( N A otherwise.
−| \ |
2Acommonterminologyforthisisstrongadditivity,whichis,however,aboutsummability
ratherthanadditivity.
3
Example 1.4. Let (S,B,µ) be a probability space. Then B A 1
A
∋ 7→ ∈
Lp(S,µ) defines a measure for 1 p< and a semi-measure for p= .
≤ ∞ ∞
LetT :V W beaboundedlinearoperatorbetweenBanachspaces. Given
a semi-measu→re (resp. measure) φ : B V, the composite map Tφ : B W
→ →
is a semi-measure (resp. measure). As a special case of this, we have a semi-
measure (resp. measure) φ : B ℓ∞(V∗) as a composition of φ with the
→ 1
canonical embedding V ℓ∞(V∗).
→ 1
b
Definition 1.5. Given a semi-measure λ : B C, the variation of λ is a
function λ :B [0, ] defined by →
| | → ∞
n
λ(A)=sup λ(A ); A is a finite partition of A in B .
j j
| | { | | { } }
j=1
X
The value λ(S) is called the total variation of λ and denoted by λ . A semi-
| | k k
measure φ is said to be of bounded variation when λ < .
k k ∞
The following are standard facts on complex (semi-)measures.
Proposition 1.6.
(i) Thevariation λ ofacomplexsemi-measureλisadditiveandsatisfiesthe
| |
inequality
sup λ(A);A B,A B λ(B) πsup λ(A);A B,A B for B B.
{| | ⊂ ∈ }≤| | ≤ {| | ⊂ ∈ } ∈
(ii) The variation of a complex measure λ is countably additive and satisfies
∞
λ(A)=sup λ(A ); A is a countable partition of A in B .
j j
| | { | | { } }
j=1
X
(iii) Anycomplexmeasuredefinedonaσ-algebraBhasafinitetotalvariation
and the vector space L1(B) of all complex measures on B is a Banach
space with the norm of total variation.
Lemma 1.7 (Half Average Inequality). For each positive d N, there exists
∈
C > 0 (C = 1, C = 1/π, C = 1/4 and so on) with the following property:
d 1 2 3
Given a finite family v Rd of euclidean vectors,we can find a finite subset
j
J 1, ,n so tha{t ∈n }v v /C .
⊂{ ··· } j=1| j|≤| j∈J j| d
Proof. WemaysupposePthatv =0. FPoraunitvectore,set(v ,e) =(v ,e) 0,
j j + j
6 ∨
which is a continuous function of e. In view of the inequality
n
v (v ,e)= (v ,e) ,
j j j +
≥
(cid:12)(vjX,e)>0 (cid:12) (vjX,e)>0 Xj=1
(cid:12) (cid:12)
(cid:12) (cid:12)
4
let e be a unit vector which maximizes the function n (v ,e) of e and set
0 j=1 j
J = j;(v ,e )>0 . Then v n (v ,e ) n (v ,e) for any
{ j 0 } | j∈J j|≥ j=1 j 0 +P≥ j=1 j +
e and have
P P P
n n
v (v ,e) de=C v with (v,e) de=C v for any v.
j j + d j + d
≥ | | | |
(cid:12)Xj∈J (cid:12) Xj=1Z|e|=1 Xj=1 Z|e|=1
(cid:12) (cid:12)
(cid:12) (cid:12)
Definition 1.8. Let φ be a V-valued semi-measure on a Boolean algebra B
with V a Banach space. The semi-variation (variation)3 of φ is a function
φ :B [0, ] ( φ :B [0, ]) defined by
| | → ∞ ||| ||| → ∞
φ(A)=sup v∗,φ (A); v∗ 1,v∗ V∗ ,
| | {|h i| k k≤ ∈ }
n
φ (A)=sup φ(A ) ; A is a finite partition of A with A B .
j j j
||| ||| { k k { } ∈ }
j=1
X
Here v∗,φ denotes a complex semi-measure v∗(φ(A)) (A B).
h i ∈
A semi-measure is said to be bounded (resp. strongly bounded) if φ
| |
(resp. φ ) is bounded. We say that φ is squeezing if lim φ(A ) = 0
n→∞ n
for any|||di|||sjoint sequence A in B|. | | |
n n≥1
{ }
Proposition 1.9.
(i) The variation of a V-valued measure is a positive measure.
(ii) A V-valued strongly bounded semi-measure φ defined on a σ-algebra is
countably additive if φ is countably additive.
||| |||
(iii) The semi-variationof a V-valued semi-measure is monotone,subadditive;
φ(A) φ(A B) φ(A) + φ(B) for A,B B, and satisfies the
| | ≤ | | ∪ ≤ | | | | ∈
inequality
sup φ(A) ;A B,A B φ(B) πsup φ(A) ;A B,A B , B B.
{k k ⊂ ∈ }≤| | ≤ {k k ⊂ ∈ } ∈
Consequently the range of a semi-measure φ is a bounded subset of V if
and only if φ is bounded, i.e., φ(S)< .
| | ∞
(iv) A semi-measure is bounded if it is squeezing. In particular, measures are
bounded.
(v) A semi-measure φ is squeezing if and only if so is φ.
| |
Proof. (i) (iii) are consequences of definitions by standard arguments.
(iv): A∼ssume that φ(B) = for some B B. Then, given any r > 0, we
can find A B in B s|uc|h that∞φ(A) r and∈ φ(B A) r. In fact, if we
⊂ k k ≥ k \ k ≥
3Warning: In literatures, semi-variation is denoted by k k, whereas | | is used to indicate
variation.
5
chooseAsothat φ(A) r+ φ(B) , φ(A) = φ(B) φ(B A) φ(B) +
k k≥ k k k k k − \ k ≤k k
φ(B A) . Byasqueezingargument,weobtainadecreasingsequence A
n n≥1
ikn B s\atiskfying φ(A ) = and φ(A ) 1+ φ(A ) for n {1. T}hus,
n n+1 n
| | ∞ k k ≥ k k ≥
the squeezing property is violated for the disjoint sequence A A .
n n+1 n≥1
{ \ }
(v): This is a consequence of (iii). The if part is trivial, whereas the only if
partis checkedas follows: If φ is notsqueezing,there existadisjoint sequence
| |
B and δ > 0 such that φ(B ) δ for n 1. Then, thanks to the π-
n n
i{neq}uality,we canfind A |B| in B≥so that φ(≥B ) π φ(A ) +1/n,which
n n n n
⊂ | | ≤ k k
denies lim φ(A ) =0.
n→∞ n
k k
Lemma 1.10.
φ(A)=sup α φ(A ) ; A is a finite partition of A with A B and α 1 with α C .
j j j j j j
| | { { } ∈ | |≤ ∈ }
(cid:13)X (cid:13)
Proof. (cid:13) (cid:13)
v∗( α φ(A )) v∗(φ(A )) = eiθjv∗(φ(A ))= v∗( eiθjφ(A )) .
j j j j j
≤ | |
(cid:12) X (cid:12) X X (cid:12) X (cid:12)
F(cid:12)(cid:12)romthefirstineq(cid:12)(cid:12)uality, v∗( αjφ(Aj)) v∗φ(A)andth(cid:12)(cid:12)en αjφ(Aj) (cid:12)(cid:12)
| |≤| | k k≤
φ(A). Fromtheequalities, v∗(φ(Aj)) eiθjφ(Aj) sup αjφ(Aj)
| | P| |≤k k≤ Pk k
and then φ(A) sup α φ(A ) .
j j
| | ≤ k P k P P
Corollary 1.11. Semi-Pvariation remains invariant under taking composition
with an isometric embedding. In particular, φ = φ. Here φ denotes the
| | | |
composition of φ with the canonical embedding V ℓ∞(V∗).
→ 1
b b
Let φ : B V be a semi-measure. For a simple function f : S C,
→ →
i.e., a function with f(S) a finite set, we note that f = 1 and
z∈f(S) [f=z]
set φ(f) = φ([f = z]). Here [f = z] = s S;f(s) = z . By
z∈f(S) { ∈ P }
subpartitioning and regrouping,the correspondence f φ(f) is linear and the
P 7→
above lemma means φ = φ(S). Therefore, if φ(S) < , φ is continuously
extended to the unifokrmkclos|u|re C(B) of the set o|f|simple∞functions. Note that
C(B) is a commutative C*-algebra. When B is a σ-algebra, C(B) is the set of
boundedmeasurablefunctionsandtheobviouspairingC(B) L1(B) Cgives
rise to inclusions C(B) L1(B)∗, L1(B) C(B)∗. × →
Conversely, any bou⊂nded linear map⊂φ : C(B) V arises in this way.
→
Thus bounded semi-measures form a Banach space with respect to the norm
φ = φ(S).
k k | |
Asequence f ofcomplex-valuedfunctionsonS issaidtoσ-converge
n n≥1
{ }
to a function f onS if f is uniformly bounded andlim f (s)=f(s) for
n n→∞ n
every s S. Let Bσ be{the}σ-algebra generated by B. Then C(Bσ) is minimal
among s∈ets which contain C(B) and have the property of being closed under
σ-convergence. A V-valued measure φ on a σ-algebra B is σ-continuous in the
sense that, if f C(B) σ-converges to f C(B), then φ(f ) φ(f) in the
n n
∈ ∈ →
weak topology.
Consider a set Λ of complex semi-measures on a Boolean algebra B and
assume that it is bounded in the sense that sup λ(S);λ Λ < . We
{| | ∈ } ∞
6
introduce then a bounded linear map φ : C(B) ℓ∞(Λ) by φ (f) : λ
Λ Λ
λ(f) C. From mutual estimates → 7→
∈
α φ (A ) = sup α λ(A ) sup λ(A)= sup δ ,φ (A) φ (A),
j Λ j j j λ Λ Λ
k k | |≤ | | |h i| ≤| |
λ∈Λ λ∈Λ λ∈Λ
X X
we see that φ (A)=sup λ(A);λ Λ for A B. In particular,
Λ
| | {| | ∈ } ∈
φ =sup λ(f);λ Λ,f C(B) =sup λ ;λ Λ <
Λ 1
k k {| | ∈ ∈ } {k k ∈ } ∞
and φ = φ . Here we set Λ = λ;λ Λ , which is again a bounded set
Λ |Λ|
| | | | | | {| | ∈ }
of semi-measures in view of the π-inequality.
Proposition 1.12. Consider the following conditions on a bounded set Λ of
complex semi-measures on a Boolean algebra B.
(i) φ is squeezing.
Λ
(ii) φ is a measure.
Λ
(iii) φ is squeezing.
|Λ|
(iv) φ is a measure.
|Λ|
(i) and (iii) are equivalent. If B is a σ-algebra and Λ consists of complex
measures, all the conditions (i) (iv) are equivalent.
∼
Proof. (ii) = (i) and (iv) = (iii) are trivial, whereas (i) follows from (iii) in
⇒ ⇒
view of λ(A) λ(A) and (iii) = (iv) is a special case of (i) = (ii).
| |≤| | ⇒ ⇒
(i) = (ii): If φ is not countably additive, we have a disjoint sequence
Λ
A in⇒B andδ >0 suchthat φ ( A ) δ for all m 1. Thenwe can
n Λ n≥m n
{ } k ⊔ k≥ ≥
inductively find a sequence λ Λ and a subsequence n < n < so that
m 1 2
∈ ···
λ ( A ) δ/2. Now the disjoint sequenceB = A
| m ⊔nm≤n<nm+1 n |≥ m ∪nm≤n<nm+1 n
satisfies φ (B ) λ (B ) δ/2 and violates the squeezing property of
Λ m m m
k k ≥ | | ≥
φ .
Λ
(i) = (iii): If φ is not squeezing, we have a disjoint sequence B in B
|Λ| n
⇒ { }
and δ > 0 such that φ (B ) δ. We can therefore find a sequence λ Λ
|Λ| n n
sothat λ (B ) δ/k2andthekn,≥bythe π-inequality,a sequenceA B i∈nB
n n n n
| | ≥ ⊂
fulfilling λ (B ) π λ (A ) +δ/3. Now the disjoint sequence A satisfies
n n n n n
| | ≤ | | { }
φ (A ) λ (A ) δ/6 and denies the squeezing property of φ .
Λ n n n Λ
k k≥| |≥
Definition 1.13. Let µ be a finite positivesemi-measureonaBoolean-algebra
B. A vector semi-measure φ on B is said to be µ-continuous if ǫ > 0, δ >
0, A B,µ(A) δ = φ(A) ǫ. ∀ ∃
∀ ∈ ≤ ⇒k k≤
Theorem 1.14 (Pettis1938). Suppose that both of φ and µ are measures (B
being a σ-algebra necessarily). Then φ is µ-continuous if and only if µ(A) = 0
(A B) implies φ(A)=0.
∈
7
Proof. We follow [DU] I.2. By taking the composition with the canonical em-
§
beddingV ℓ∞(V∗),wemaysupposethatφ=φ ,whereΛ= v∗φ;v∗ V∗
is a bounde→d subset1of L1(B). Λ { ∈ 1}
If φ is not µ-continuous, there exists δ > 0 and a sequence A B such
n
that φ(A ) δ for n 1 and ∞ µ(A ) < . Let B =∈ A
be a kdecrenasking≥sequence ≥in B. Fromn=th1e latnter ine∞quality, µ(mB ) ∪n0≥,mi.e.n,
m
P ↓
µ( B )=0. From the former inequality, we have
m
∩
φ (B ) =sup v∗φ(B );v∗ V∗ sup v∗φ(A );v∗ V∗
k |Λ| m k {| | m ∈ 1}≥ {| | m ∈ 1}
sup v∗,φ(A ) ;v∗ V∗ = φ(A ) δ.
≥ {|h m i| ∈ 1} k m k≥
Since φ is a measure, φ is also a measure by Proposition 1.12 and the limit
Λ |Λ|
m is applied to get φ ( B ) δ. Therefore we can find a functional
|Λ| m
v∗→∞V∗ such that v∗φk( B ∩) > δk/2≥and then A B in B such that
∈ 1 | | ∩ m ⊂ ∩ m
π v∗,φ(A) >δ/2. Thus φ(A) =0, whereas µ(A) µ( B )=0.
m
|h i| k k6 ≤ ∩
Theorem 1.15 (Doubrovsky1947). Let Λ be a bounded set of complex mea-
sures on a σ-algebra B. If φ :B ℓ∞(Λ) is a measure, there exists a positive
Λ
measure µ L1(B) for which φ→is µ-continuous with a reverse inequality
Λ
∈
µ(A) φ (A)=sup λ(A);λ Λ .
Λ
≤| | {| | ∈ }
Proof. This is [DH], Theorem I.2.4. We first establish a kind of compactness of
a bounded Λ: Given any ǫ > 0, we can find a finite subset F Λ such that if
A B satisfies λ(A)=0 for λ F, then λ(A) ǫ for any λ⊂ Λ.
∈Ifnot,there|ex|istsδ >0such∈thatforan|y|F ⋐≤Λ,wecanfind∈A Bsatisfy-
∈
ing λ(A)=0foranyλ F but ν (A) δ forsomeν Λ. Thenwecaninduc-
tive|ly|choose a sequence∈λ and|A| B≥so that λ (A∈)= = λ (A )=0
n n 1 n n n
∈ | | ··· | |
and λ (A ) δ for n 1. Let B = A be a decreasing sequence
n+1 n m n≥m n
| | ≥ ≥ ∪
and set B = B . Since λ (B ) λ (B ) λ (A ) = 0 for
∞ ∩n n | m| ∞ ≤ | m| m ≤ n≥m| m| n
m 1 and lim sup λ(B B )=0, we have
≥ m→∞ λ∈Λ| | m\ ∞ P
0= lim λ (B B )= lim λ (B ),
m+1 m ∞ m+1 m
m→∞| | \ m→∞| |
which contradicts with λ (B ) λ (A ) δ.
m+1 m m+1 m
| | ≥| | ≥
Now we use the boundedness of Λ again to construct a control measure µ
over Λ. For each n 1, choose F ⋑ Λ so that λ(A) = 0 with A B
≥ n λ∈Fn| | ∈
implies λ(A) 1/nforanyλ Λ. Thenthe positivemeasureµ = λ
satisfies|µ| (A)≤ F φ (A) fo∈r A B and, if wPe define n λ∈Fn| |
n n Λ
≤| || | ∈ P
∞
1
µ= µ ,
F 2n n
n
n=1| |
X
it is a positive finite measure on B with the property µ(A) φ (A). Assume
Λ
≤| |
that µ(A) = 0. Then from µ (A) = 0, λ(A) 1/n for any λ Λ and
n
| | ≤ ∈
any n 1, i.e., λ(A) = 0. Thanks to the Pettis theorem, this means the
≥ | |
µ-continuity of φ .
Λ
8
Corollary 1.16 (Bartle-Dunford-Schwartz1955). For a vector measure φ on a
σ-algebra,we can find a finite positive measure µ so that φ is µ-continuousand
µ is majorized by φ.
| |
Proof. Apply the theorem to Λ= v∗φ;v∗ V∗ , which is bounded by Propo-
{ ∈ 1}
sition 1.9 (iv).
Recall that countable additivity of a semi-measure µ: B [0, ) is equiv-
alent to the condition that, if A in B, then µ(A ) 0→. Le∞t Bσ be the
n n
σ-algebrageneratedbyB. Theclass↓ic∅alextensiontheorem4↓saysthat,ifafinite
positive semi-measure on B is countably additive, it is uniquely extended to a
positive measure on Bσ.
In the framework of Daniell integral (see [12] for example), this can be ex-
plainedinthefollowingfashion: LetLbethevectorlatticeofreal-valuedsimple
functions on the base set S. Then a semi-measure µ can be interpreted as a
positive linear functional L R, which is also denoted by µ. Let f 0 in L.
n
Then, given ǫ > 0, [f ǫ]→ in B and µ(f ) f µ([f ǫ])↓+ǫµ(S),
n n 1 ∞ n
≥ ↓ ∅ ≤ k k ≥
together with the continuity of µ imply lim µ(f ) ǫµ(S). Thus, µ is contin-
n n
≤
uous as a linear functional and we can apply the whole construction of Daniell
integral to get a measure extension to Bσ.
Lemma 1.17. Let µ be a finite positive measure on Bσ and embed Bσ into
L1(S,µ). Then Bσ is closed in L1(S,µ) and B is dense in Bσ.
Proof. Sinceanysequentialconvergenceinmeanimpliesalmostallconvergence
by passing to a subsequence, Bσ (more precisely 1 ;B Bσ ) is closed in
B
{ ∈ }
L1(S,µ)(pointwiseconvergenceof 0,1 -valuedfunctionsproduce 0,1 -valued
{ } { }
functions). In view of 1 = 1 1 and the dominated convergence theorem,
A∩B A B
onseesthattheclosureofB(moreprecisely 1 ;B sB )inL1(S,µ)provides
B
a σ-algebra and hence coincides with Bσ. { ∈ }
Theorem 1.18 (Kluv´anek1961). Let φ : B V be a semi-measure on a
Boolean algebra B. If φ is µ-continuous for so→me countably additive positive
semi-measure µ on B, then φ is uniquely extended to a measure Bσ V on
the σ-algebra Bσ generated by B. →
Proof. The uniqueness is as usual: Given two extensions, sets of their coinci-
dency form a σ-algebra containing B, whence extensions coincide on the whole
Bσ.
For the existence, first note that µ is extended to a measureby the classical
extension theorem, which is again denoted by µ. From the previous lemma,
a complete (pseudo)metric on Bσ is defined by d(A,B) = 1 1 =
A B 1
µ(A B) = µ(A B)+µ(B A) so that (Bσ,d) is the complketion−of (Bk,d).
△ \ \
In view of the inequality
φ(A) φ(B) = φ(A B) φ(B A) φ(A B) + φ(B A) ,
k − k k \ − \ k≤k \ k k \ k
4This can be attributed to many researchers: Fr´echet, Carath´eodory, Kolmogorov, Hahn
andHopf(VladimirI.Bogachev,MeasureTheoryI).ItseemsfairtoaddthenameofDaniell
tothelistbecausethetheorem itselfisjustanexampleofDaniellintegral.
9
φisuniformlycontinuouswithrespecttod,whichadmitsthereforeacontinuous
extension φ to (Bσ,d).
Now φ is countably additive thanks to the d-continuity: finite additivity of
φgoesovertothatofφandmonotoneconvergenceassurestheσ-additivity.
2 Cross Norms
This is a veryoldbut still developing subjectandthere arelots of referencesto
be mentioned. We nominate, however, just [Raymond 1973] and [Diesel 1985]
here to follow them.
GivenBanachspacesX andY,letB(X,Y)betheBanachspaceofbounded
bilinear forms on X Y and L(X,Y) be the Banach space of bounded linear
maps of X into Y. ×There are natural identifications L(X,Y∗) = B(X,Y) =
L(Y,X∗). Recall that a (semi)norm on the algebraic tensor product X Y is
⊗
called a cross (semi)norm if it satisfies x y = x y .
The obvious bilinear map X Y k B⊗(Xk∗,Yk∗)kgkiveks rise to a linear map
X Y B(X∗,Y∗) by (x y)×(x∗,y→∗) = x∗(x)y∗(y), which is injective. In
⊗ → ⊗
fact, let z = x y X Y and express z = z e f
l l ⊗ l ∈ ⊗ 1≤j≤m,1≤k≤n j,k j ⊗ k
with e X and f Y linearly independent. The dual bases e∗ and
f∗ {arje}c⊂ontPinuous{onk}fi⊂nite-dimensional subspacesPand can be ext{enjd}ed to
{ k}
bounded linear functionals. We then have z = z(e∗,f∗). Thus the norm on
j,k j k
B(X∗,Y∗) induces a cross norm on X Y, which is denoted by B(X∗,Y∗).
In the embedding X Y B(X∗,Y∗)⊗= L(X∗,Y∗∗), x yk·kX Y is
⊗ → l l ⊗ l ∈ ⊗
realized by the operator x∗ x∗(x )y and the image of X Y is included
in the subspace L(X∗,Y) 7→L(Xl∗,Y∗∗l). lThus we have P ⊗
⊂ P
k xl⊗ylkB(X∗,Y∗) =sup{k x∗(xl)ylk;x∗ ∈X1∗}=sup{| x∗(xl)y∗(yl)|;x∗ ∈X1∗,y∗ ∈Y1∗}
l l l
X X X
and a similar expression holds with the role of X and Y exchanged.
There is another natural way to get a cross norm on X Y. Each f
B(X,Y)definesalinearfunctionalonX Y byf(z)= n f(⊗x ,y )satisfyin∈g
f(z) n f x y , whence it ind⊗uces a linear mapl=X1 Yl l B(X,Y)∗
| |≤ l=1k kk lkk lk P ⊗ →
and the associated seminorm B(X,Y)∗ satisfies
P k·k
n n
z B(X,Y)∗ =inf xl yl ;z = xl yl .
k k { k kk k ⊗ }
l=1 l=1
X X
The inequality is clear. To get the reverse inequality, we first notice that the
right hand side≤defines a seminorm on X Y. Let ϕ:X Y C be a
inf
k·k ⊗ ⊗ →
-bounded linear functional with its dual norm denoted by ϕ . Then the
inf
k·k k k
associated bilinear functional f(x,y)=ϕ(x y) satisfies f ϕ , whence
⊗ k k≤k k
z =sup ϕ(z); ϕ 1 sup f(z);f B(X,Y), f 1 = z B(X,Y)∗.
k k {| | k k≤ }≤ {| | ∈ k k≤ } k k
The bilinear map Φ:X∗ Y∗ B(X,Y) defined by
× →
Φ(x∗,y∗):(x,y) x∗(x)y∗(y)
7→
10