On vector measures and extensions of transfunctions

0.15 MB·

by  Piotr Mikusinski

ON VECTOR MEASURES AND EXTENSIONS OF
TRANSFUNCTIONS
7
1 PIOTR MIKUSIN´SKI1 and JOHN PAUL WARD2
0
2
Abstract. We are interested in extending operators defined on
n
a positive measures, called here transfunctions, to signed measures
J and vector measures. Our methods use a somewhat nonstandard

7 approach to measures and vector measures. The necessary back-
1 ground, including proofs of some auxiliary results, is included.

]
A
F
1. Introduction
.
h
t By a transfunction [3] we mean a map between sets of measures on
a
m measurable spaces. More precisely, if (X,Σ ) and (Y,Σ ) are measur-
X Y
[ able spaces and M(X,Σ ,R+) and M(Y,Σ ,R+) are the sets of all
X Y
1 finite positive measures on ΣX and on ΣY, respectively, by a transfunc-
v tion from (X,Σ ) to (Y,Σ ) we mean a map
X Y
7
3 Φ : M(X,Σ ,R+) → M(Y,Σ ,R+).
X Y
8
4 Ifnecessary, atransfunctioncanbedefinedonasubsetofM(X,Σ ,R+).
X
0 If f : (X,Σ ) → (Y,Σ ) is a measurable function, then
. X Y
1
Φ (µ)(B) = µ(f−1(B)),
0 f
7 for µ ∈ M(X,Σ ,R+) and B ∈ Σ , defines a transfunction from
1 X Y
: M(X,Σ ,R+) to M(X,Σ ,R+). If a transfunction Φ is of this form,
v X Y
i then clearly the function can be reconstructed from the action of Φ.
X
If X = Y = [0,1], Σ is the σ-algebra of Lebesgue measurable subsets
r
a of [0,1], and λ is the Lebesgue measure on [0,1], then
Φ(µ) = µ([0,1])λ
defines a transfunction that does not correspond to a function. It can
be thought of as a “map” that distributes every input uniformly on
[0,1].
We are interested in transfunctions as a generalization of a function
from X to Y. Instead of mapping a point x ∈ X to a point y ∈ Y a
2010 Mathematics Subject Classification. Primary 46G10; Secondary 28B05,
46B22.
Key words and phrases. Vector measures, transfunctions, Radon-Nikodym
property.
1
2 P. MIKUSIN´SKIand J.P. WARD
transfunction can be thought of as mapping a probability distribution
of the input to a probability distribution of the output. Our long term
goal is to investigate to what extent the tools developed for functions
can be extended to transfunctions.
We consider the following properties of transfunctions:
Weakly additive: Φ(µ +µ ) = Φ(µ )+Φ(µ )forallµ ,µ ∈ M
1 2 1 2 1 2
such that µ ⊥ µ ,
1 2
Strongly additive: Φ(µ +µ ) = Φ(µ )+Φ(µ ) for all µ ,µ ∈
1 2 1 2 1 2
M,
Homogeneous: Φ(αµ) = αΦ(µ) for any α > 0,
Monotone: Φ(µ ) ≤ Φ(µ ), if µ ≤ µ ,
1 2 1 2
Norm preserving: kΦ(µ)k = kµk,
Bounded: kΦ(µ)k ≤ Ckµk for some C > 0 and all µ ∈ M,
Continuous with respect to weak convergence: Ifµ (A) →
n
µ(A) for every A ∈ Σ , then Φ(µ )(B) → Φ(µ)(B) for every
X n
B ∈ Σ .
Y
Continuous with respect to uniform convergence: Ifkµ −
n
µk → 0, then kΦ(µ )−Φ(µ)k → 0.
n
The definition of transfunctions is very general and it makes sense
if finite positive measures are replaced by signed measures of bounded
variation or by vector valued measures of bounded variation. More-
over, properties of transfunctions listed above, with the exception of
monotonicity, make sense in this general setting. In the last section of
this paper we define extensions of transfunctions to signed measures
and vector measures. We also discuss the question of uniqueness of
such extensions.
The next three sections contain the necessary background and proofs
of some auxiliary results.
2. Measure spaces
In this section we recall some definitions and results from [2]. While
the results are standard, the approach in [2] is different.
Let X be a nonempty set. A collection R of subsets of X is called
a ring of subsets of X if A,B ∈ R implies A∪B,A\B ∈ R. A map
µ : R → [0,∞) is called σ-additive if for any sequence of disjoint sets
∞
A ,A ,··· ∈ R such that A ∈ R we have
1 2 n=1 n
S∞ ∞
µ A = µ(A ).
n n
!
n=1 n=1
[ X
ON VECTOR MEASURES AND EXTENSIONS OF TRANSFUNCTIONS 3
In this note we use the same symbol to denote a subset of X and the
characteristic function of that set, that is, if A ⊂ X we will write
1 if x ∈ A
A(x) = .
0 otherwise
(cid:26)
Let µ beaσ-additive measure ona ringR ofsubsets ofX. Aset S ⊂
X is called µ-summable if there are A ,A ,··· ∈ R and α ,α ,··· ∈
1 2 1 2
{−1,1} such that
∞
I µ(A ) < ∞;
n=1 n
∞ ∞
II S(x) = α A (x)foreveryx ∈ X forwhich A (x) <
n=1 n n n=1 n
P
∞.
P ∞ P
If conditions I and II are satisfied, we write S ≃ α A .
n=1 n n
∞ ∞
It can be shown that, if S ≃ α A and S ≃ β B ,
n=1 n n n=1 n n
∞ ∞ P
then α µ(A ) = β µ(B ). This enables us to define an
n=1 n n n=1 n n
P P
extension of µ to all summable sets:
P ∞ P ∞
If S ≃ α A , then µ(S) = α µ(A ).
n=1 n n n=1 n n
A set S ⊂ X is called µ-measurable if the set S ∩A is µ-summable
P P
for every A ∈ R. Let Σ(R,µ) denote the collection of all µ-measurable
subsets of X. If S ⊂ X is a µ-measurable set that is not µ-summable,
then we define µ(S) = ∞. It can be shown that Σ(R,µ) is a σ-algebra
containing R and (X,Σ(R,µ),µ) is a complete measure space.
The followinglemma will beneeded intheproofofaresult inthelast
section. We denote by µ the restriction of µ to A, that is, µ (S) =
A A
µ(S ∩A).
Lemma 2.1. Let µ ,...,µ be finite measures on (X,Σ). Then there
1 n
exists a measure µ on (X,Σ) such that for every ε > 0 there is a finite
partition π of X such that for i = 1,...,n we have
µ = α µ +κ
i i,S S i
S∈π
X
where α ≥ 0 and κ ,...,κ are measures on (X,Σ) such that
i,S 1 n
(2.1) κ (X)+···+κ (X) < ε.
1 n
In particular, we may define µ to be n µ .
i=1 i
Proof. Let ε > 0. Notice that the thPe measures µ1,...,µn are abso-
lutely continuous with respect to µ as defined above. Consider the
Radon-Nikodym derivatives f of µ , that is,
i i
µ (B) = f dµ
i i
ZB
for all B ∈ Σ. Since each f is a non-negative and integrable with
i
respect to µ, there are simple functions α A(x), with respect
A∈πi i,A
P
4 P. MIKUSIN´SKIand J.P. WARD
to finite partitions π of X, such that α ≥ 0, α A(x) ≤ f ,
i i,A A∈πi i,A i
and
P
ε
f − α A(x) dµ < .
i i,A
n
ZX AX∈πi !
Now define the common refinement of the partitions π to be π, and
i
define the measures κ by the equation
i
κ (B) = f − α dµ
i i i,A
ZB AX∈πi !
for all B ∈ Σ. Notice that each simple function with respect to π is
i
also a simple function with respect to π, that is,
α A(x) = α S(x)
i,A i,S
AX∈πi XS∈π
where α = α if S ⊆ A. Consequently, for every B ∈ Σ, we have
i,A i,S
µ (B) = α S(x)dµ+ f − α S(x) dµ
i i,S i i,S
ZB S∈π ZB S∈π !
X X
= α µ (B)+κ (B),
i,S S i
S∈π
X
and the κ ’s were constructed to satisfy (2.1). (cid:3)
i
3. Vector Measures
Let X be a nonempty set, R a ring of subsets of X, and let (E,k·k)
be a Banach space.
A set function µ : R → E is called σ-additive if for any sequence of
∞
disjoint sets A ,A ,··· ∈ R such that A ∈ R we have
1 2 n=1 n
∞ ∞
S
µ A = µ(A ).
n n
!
n=1 n=1
[ X
Definition 3.1. By the variation of a σ-additive set function µ : R →
E we mean the set function |µ| : R → [0,∞] defined by
|µ|(A) = sup ||µ(B)|| : π ⊂ R is a finite partition of A .
( )
B∈π
X
Note that |µ|(A) ≥ kµ(A)k for any set A ∈ R. If |µ|(X) < ∞, then we
say that µ is of bounded variation.
In the remainder of this section we assume that µ : R → E is a
σ-additive set function of bounded variation.
ON VECTOR MEASURES AND EXTENSIONS OF TRANSFUNCTIONS 5
Lemma 3.2. |µ| is monotone and finitely additive.
Proof. Clearly |µ| is monotone. Now consider disjoint A ,...,A ∈ R.
1 n
Since the union of finite partitions of A ,...,A is a finite partition of
1 n
A ∪···∪A , we have
1 n
n n
|µ| A ≥ |µ|(A ).
k k
!
k=1 k=1
[ X
On the other hand, every finite partition π of n A can be refined to
k=1 k
a partition n π , where π is a finite partitionof A for k = 1,...,n.
k=1 k k k
S
Then
S
n n
||µ(B)|| ≤ ||µ(B)|| = ||µ(B)|| ≤ |µ|(A )
k
BX∈π B∈πX1∪···∪πn Xk=1BX∈πk Xk=1
and consequently
n n
|µ| A ≤ |µ|(A ).
k k
!
k=1 k=1
[ X
(cid:3)
Theorem 3.3. |µ| is a σ-additive positive measure on R.
Proof. Let A ,A ,··· ∈ R be a sequence of disjoint sets such that
1 2
A = ∞ A ∈ R. By Lemma 3.2, for every m ∈ N, we have
n=1 n
S m
|µ|(A ) ≤ |µ|(A).
n
n=1
X
Consequently
∞
|µ|(A ) ≤ |µ|(A).
n
n=1
X
Now let ε be an arbitrary positive number. Then
k
|µ|(A) < kµ(B )k+ε,
j
j=1
X
6 P. MIKUSIN´SKIand J.P. WARD
for some partition {B ,...,B } of A, and we have
1 k
k
|µ|(A) < kµ(B )k+ε
j
j=1
X
k ∞
= µ (A ∩B ) +ε
n j
(cid:13) !(cid:13)
Xj=1 (cid:13) n[=1 (cid:13)
(cid:13) (cid:13)
k (cid:13)∞ (cid:13)
≤ (cid:13) kµ(A ∩B )k+(cid:13)ε
n j
j=1 n=1
XX
∞ k
= kµ(A ∩B )k+ε
n j
n=1 j=1
XX
∞
≤ |µ(A )|+ε.
n
n=1
X
Since ε is arbitrary, we obtain
∞
|µ|(A) ≤ |µ(A )|.
n
n=1
X
(cid:3)
Corollary 3.4. |µ| can be extended to a complete σ-additive measure
on Σ(R,|µ|).
We are going to use |µ| to denote the variation of µ defined on R as
well as its extension to a positive measure on Σ(R,|µ|).
Our next goal is to define an extension of µ to a σ-additive E-valued
measure on Σ(R,|µ|).
∞
Lemma 3.5. If ∅ ≃ α A for some A ∈ R and α ∈ {−1,1},
n n n n
n=1
∞
P
then α µ(A ) = 0.
n n
n=1
P
∞ ∞
Proof. First note that, if ∅ ≃ α A , then α |µ|(A ) = 0. Let
n n n n
n=1 n=1
ε > 0 and let n0 ∈ N be such thPat P
∞
|µ|(A ) < ε.
n
n=Xn0+1
ON VECTOR MEASURES AND EXTENSIONS OF TRANSFUNCTIONS 7
Let C ,...,C ,D ,...,D ∈ R be such that C ∩ D = ∅ for all j
1 n1 1 n2 j k
and k and such that
(3.1) α A +···+α A = C +···+C −D −···−D .
1 1 n0 n0 1 n1 1 n2
Note that
∅ ≃ C +···+C −D −···−D +α A +α A +...
1 n1 1 n2 n0+1 n0+1 n0+2 n0+2
If V = supp(D ∪···∪D ), then
1 n2
∅ ≃ −D −···−D +α (A ∩V)+α (A ∩V)+...
1 n2 n0+1 n0+1 n0+2 n0+2
and hence
0 = −|µ|(D )−···−|µ|(D )+α |µ|(A ∩V)+α |µ|(A ∩V)+...
1 n2 n0+1 n0+1 n0+2 n0+2
< −|µ|(D )−···−|µ|(D )+ε.
1 n2
Consequently
(3.2) |µ|(D )+···+|µ|(D ) < ε
1 n2
and since
n1 n2 ∞
0 = |µ|(C )− |µ|(D )+ α |µ|(A ),
n n n n
n=1 n=1 n=n0+1
X X X
also
(3.3) |µ|(C )+···+|µ|(C ) < 2ε.
1 n2
From (3.1), (3.2), and (3.3) we obtain
n0
α µ(A ) = kµ(C )+···+µ(C )−µ(D )−···−µ(D )k
n n 1 n1 1 n2
(cid:13) (cid:13)
(cid:13)Xn=1 (cid:13)
(cid:13) (cid:13) ≤ kµ(C )k+···+kµ(C )k+kµ(D )k+···+kµ(D )k
(cid:13) (cid:13) 1 n1 1 n2
(cid:13) (cid:13)
≤ |µ|(C )+···+|µ|(C )+|µ|(D )+···+|µ|(D )
1 n1 1 n2
< 3ε
For any m > n we have
0
m n0 m
α µ(A ) ≤ α µ(A ) + α µ(A )
n n n n n n
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)Xn=1 (cid:13) (cid:13)Xn=1 (cid:13) (cid:13)n=Xn0+1 (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) n0 (cid:13) (cid:13) m (cid:13)
(cid:13) (cid:13) ≤ (cid:13) α µ(A )(cid:13)+(cid:13) kµ(A )k (cid:13)
n n n
(cid:13) (cid:13)
(cid:13)Xn=1 (cid:13) n=Xn0+1
(cid:13)m (cid:13)
(cid:13) (cid:13)
≤ (cid:13) |µ|(A ) <(cid:13)4ε
n
n=1
X
8 P. MIKUSIN´SKIand J.P. WARD
∞
Since ε is arbitrary, we obtain α µ(A ) = 0. (cid:3)
n n
n=1
P
∞ ∞
Corollary 3.6. If S ≃ α A and S ≃ β B for some A ,B ∈
n n n n n n
n=1 n=1
∞ ∞
P P
R and α ,β ∈ {−1,1}, then α µ(A ) = β µ(B ).
n n n n n n
n=1 n=1
P P
Proof. It suffices to note that
∅ ≃ α A −β B +α A −β B +...
1 1 1 1 2 2 2 2
(cid:3)
and then use Lemma 3.5.
TheabovecorollaryjustifiesthefollowingdefinitionofE-valuedmea-
sure on Σ(R,|µ|).
∞
Definition 3.7. If S ≃ α A for some A ∈ R and α ∈ {−1,1},
n n n n
n=1
∞
P
then we define µ(S) = α µ(A ).
n n
n=1
P
It remains to show that µ is σ-additive on Σ(R,|µ|). If U ⊂ X,
U ,U ,··· ∈ Σ(R,|µ|), and
1 2
∞
I |µ|(U ) < ∞;
n=1 n
∞ ∞
II U(x) = α U (x) forevery x ∈ X forwhich U (x) <
n=1 n n n=1 n
P
∞,
P P
∞
then we write U ≃ α U .
n n
n=1
P
∞
Lemma 3.8. If U ≃ α U for some U ∈ Σ(R,|µ|) and α ∈
n n n n
n=1
∞
P
{−1,1}, then µ(U) = α µ(U ).
n n
n=1
P
Proof. Since U ,U ,··· ∈ Σ(R,|µ|), for every k ∈ N there exists an
1 2
∞
expansion U ≃ β B , with B ∈ R and such that
k k,n k,n k,n
n=1
P
∞
|µ|(B ) < |µ|(U )+2−k.
k,n k
n=1
X
ON VECTOR MEASURES AND EXTENSIONS OF TRANSFUNCTIONS 9
∞
∞
Let γ C beaseriesarrangedfromallseries α β B . Then
n n n=1 k k,n k,n
n=1
∞
P P
U ≃ γ C and
n n
n=1
P ∞ ∞
µ(U) = γ µ(C ) = α µ(U ).
n n n n
n=1 n=1
X X
(cid:3)
Theorem 3.9. µ is σ-additive on Σ(R,|µ|).
∞ ∞
Proof. If U ,U ,··· ∈ Σ(R,|µ|) are disjoint, then U ≃ U .
1 2 n=1 n n=1 n
Therefore µ( ∞ U ) = ∞ µ(U ), by Lemma 3.8. (cid:3)
n=1 n n=1 n
S P
4. VectorSmeasuresPand Bochner integrable functions
Let (X,Σ,µ) be a measure space and let E be a Banach space.
A function f : X → E is Bochner integrable [4] if there are sets
A ,A ,··· ∈ Σ and vectors v ,v ,··· ∈ E such that
1 2 1 2
∞
I µ(A ) < ∞;
n=1 n
∞ ∞
II f(x) = v A (x)foreveryx ∈ X forwhich kv kA (x) <
n=1 n n n=1 n n
P
∞.
P P
If for some f : X → E, A ,A ,··· ∈ Σ, and v ,v ,··· ∈ E conditions I
1 2 1 2
∞
and II are satisfied, we will write f ≃ v A .
n=1 n n
If f : X → E is a Bochner integrable function on a measure space
P
(X,Σ,µ), then
ω(S) = fdµ
ZS
defines a vector measure of bounded variation on (X,Σ). Not every
vector measure is of this form (see [1]). We say that a Banach space
E has the Radon-Nikodym property if for every vector measure ω on
(X,Σ) of bounded variation that is absolutely continuous with respect
to a measure µ on (X,Σ) there exists a Bochner integrable function f
on (X,Σ,µ) such that ω(S) = fdµ for every S ∈ Σ.
S
If µ ,µ ,... are positive measures on a measurable space (X,Σ) and
1 2
v ,v ,··· ∈ E are such that R
1 2
∞
kv kµ (X) < ∞,
n n
n=1
X
then
∞
ω(S) = v µ (S)
n n
n=1
X
10 P. MIKUSIN´SKIand J.P. WARD
defines a vector measure of bounded variation on (X,Σ).
Theorem 4.1. If E has the Radon-Nikodym property, then every E-
valued measure ω on a measure space (X,Σ) is of the form
∞
ω(S) = v µ (S)
n n
n=1
X
where µ ,µ ,... are positive measures on (X,Σ) and v ,v ,··· ∈ E are
1 2 1 2
∞
such that kv kµ (X) < ∞.
n=1 n n
Proof. LetPω be a E-valued measure of bounded variation on (X,Σ). If
E has the Radon-Nikodym property, there exists a Bochner integrable
function f on (X,Σ,|ω|) such that ω(S) = fd|ω| for every S ∈ Σ.
S
Let A ,A ,··· ∈ Σ and v ,v ,··· ∈ E be such that f ≃ ∞ v A .
1 2 1 2 R n=1 n n
Then
∞ P
fd|ω| = v |ω|(S ∩A ).
n n
ZS n=1
X
If we define
µ (S) = |ω|(S ∩A ),
n n
then
∞
ω(S) = fd|ω| = v µ (S).
n n
ZS n=1
X
(cid:3)
5. Extension of transfunctions
In this section we define extensions of transfunctions to signed mea-
sures and vector measures. We are interested in properties of such
extensions and the question of uniqueness.
By M(X,Σ,R) we denote the collection of all signed measures on
(X,Σ) with bounded variation. For every µ ∈ M(X,Σ,R) the Jordan
decompositiontheoremdefinesuniquemeasuresµ+,µ− ∈ M(X,Σ,R+)
such that µ = µ+ − µ− and µ+ ⊥ µ−. If Φ : M(X,Σ ,R+) →
X
M(Y,Σ ,R+) is a transfunction, then Φ has a natural extension to a
Y
transfunction Φ˜ : M(X,Σ ,R) → M(Y,Σ ,R), namely
X Y
Φ˜µ = Φµ+ −Φµ−.
Clearly, there are other ways of extending Φ to a transfunction Φ˜ :
M(X,Σ ,R) → M(Y,Σ ,R). We show that the above extension has
X Y
desirable properties and that under some additional conditions it is
unique.