P-adic Measures and P-adic Spaces of Continuous Functions

P-adic Measures and P-adic Spaces of Continuous Functions

A. K. Katsaras

Department of Mathematics, University of Ioannina 45110 Ioannina, Greece

akatsar@uoi.gr

Received: 8.5.2007; accepted: 2.10.2008.

Abstract. For X a Hausdorff zero-dimensional topological space and E a Hausdorff non- Archimedean locally convex space, let C(X, E) (resp. C

(X, E)) be the space of all continuous (resp. bounded continuous ) E-valued functions on X. Some of the properties of the spaces C(X, E), C

(X, E), equipped with certain locally convex topologies, are studied. Also, some complete spaces of measures, on the algebra of all clopen subsets of X, are investigated.

Keywords: Non-Archimedean fields, zero-dimensional spaces, Banaschewski compactifica- tion, locally convex spaces

MSC 2000 classification: 46S10, 46G10

Introduction

Let K be a complete non-Archimedean valued field and let C(X, E) be the space of all con- tinuous functions from a zero-dimensional Hausdorff topological space X to a non-Archimedean Hausdorff locally convex space E. We will denote by C

(X, E) (resp. by C

(X, E)) the space of all f ∈ C(X, E) for which f (X) is a bounded (resp. relatively compact) subset of E. The dual space of C

(X, E), under the topology t

of uniform convergence, is a space M (X, E

) of finitely-additive E

-valued measures on the algebra K(X) of all clopen , i.e. both closed and open, subsets of X. Some subspaces of M (X, E

) turn out to be the duals of C(X, E) or of C

(X, E) under certain locally convex topologies.

In section 2 of this paper, we study some of the properties of the so called Q-integrals, a concept given by the author in [14]. In section 3, we identify the dual of C

(X, E) under the strict topology β

. In section 4, we prove that the dual space of C(X, E), under the topology of uniform convergence on the bounding subsets of X, is the space of all m ∈ M (X, E

) which have a bounding support. In section 5 it is shown that the space M

(X) of all separable members of M (X), under the topology of uniform convergence on the uniformly bounded equicontinuous subsets of C

(X), is complete. The same is proved in section 6 for the space M

(X) of those separable m for which the support of the extension m

, to all of the Banaschewski compactification β

X of X, is contained in the N-repletion υ

X of X, if we equip M

(X) with the topology of uniform convergence on the pointwise bounded equicontinuous subsets of C(X).

http://siba-ese.unisalento.it/ c 2010 Universit` a del Salento

1 Preliminaries

Throughout this paper, K will be a complete non-Archimedean valued field, whose valua- tion is non-trivial. By a seminorm, on a vector space over K, we will mean a non-Archimedean seminorm. Similarly, by a locally convex space we will mean a non-Archimedean locally convex space over K (see [25]). Unless it is stated explicitly otherwise, X will be a Hausdorff zero- dimensional topological space , E a Hasusdorff locally convex space and cs(E) the set of all continuous seminorms on E. The space of all K-valued linear maps on E is denoted by E

, while E

denotes the topological dual of E. A seminorm p, on a vector space G over K, is called polar if p = sup{|f | : f ∈ G

, |f | ≤ p}. A locally convex space G is called polar if its topology is generated by a family of polar seminorms. A subset A of G is called absolutely convex if λx + µy ∈ A whenever x, y ∈ A and λ, µ ∈ K, with |λ|, |µ| ≤ 1. We will denote by β

X the Banaschewski compactification of X (see [5]) and by υ

X the N-repletion of X, where N is the set of natural numbers. We will let C(X, E) denote the space of all continuous E-valued functions on X and C

(X, E) (resp. C

(X, E)) the space of all f ∈ C(X, E) for which f (X) is a bounded (resp. relatively compact) subset of E. In case E = K, we will sim- ply write C(X), C

(X) and C

(X) respectively. For A ⊂ X, we denote by χ

the K-valued characteristic function of A. Also, for X ⊂ Y ⊂ β

X, we denote by ¯ B

the closure of B in Y . If f ∈ E

, p a seminorm on E and A ⊂ X, we define

kf k

= sup

x∈X

p(f (x)), kf k

= sup

x∈A

p(f (x)).

The strict topology β

on C

(X, E) (see [9]) is the locally convex topology generated by the seminorms f 7→ khf k

, where p ∈ cs(E) and h is in the space B

(X) of all bounded K-valued functions on X which vanish at infinity, i.e. for every ǫ > 0 there exists a compact subset Y of X such that |h(x)| < ǫ if x / ∈ Y .

Let Ω = Ω(X) be the family of all compact subsets of β

X \ X. For H ∈ Ω, let C

be the space of all h ∈ C

(X) for which the continuous extension h

to all of β

X vanishes on H. For p ∈ cs(E), let β

be the locally convex topology on C

(X, E) generated by the seminorms f 7→ khf k

, h ∈ C

. For H ∈ Ω, β

is the locally convex topology on C

(X, E) generated by the seminorms f 7→ khf k

, h ∈ C

, p ∈ cs(E). The inductive limit of the topologies β

, H ∈ Ω, is the topology β. Replacing Ω by the family Ω

of all K-zero subsets of β

X, which are disjoint from X, we get the topology β

. Recall that a K-zero subset of β

X is a set of the form {x ∈ β

X : g(x) = 0}, for some g ∈ C(β

X). We get the topologies β

and β

replacing Ω by the family Ω

of all Q ∈ Ω with the following property: There exists a clopen partition (A

)

of X such that Q is disjoint from each A

βoX

. Now β

is the inductive limit of the topologies β

, Q ∈ Ω

. The inductive limit of the topologies β

, as H ranges over Ω

, is denoted by β

, while β

is the projective limit of the topologies β

, p ∈ cs(E).

For the definition of the topology β

on C

(X) we refer to [12].

Let now K(X) be the algebra of all clopen subsets of X. We denote by M (X, E

) the space of all finitely-additive E

-additive measures m on K(X) for which the set m(K(X)) is an equicontinuous subset of E

. For each such m, there exists a p ∈ cs(E) such that kmk

= m

(X) < ∞, where, for A ∈ K(X),

m

(A) = sup{|m(B)s|/p(s) : p(s) 6= 0, A ⊃ B ∈ K(X)}.

The space of all m ∈ M (X, E

) for which m

(X) < ∞ is denoted by M

(X, E

). For m ∈ M

(X, E

) we define N

on X by

N

(x) = inf{m

(V ) : x ∈ V ∈ K(X)}.

In case E = K, we denote by M(X) the space of all finitely-additive bounded K-valued measures on K(X). An element m of M (X) is called τ -additive if m(V

) → 0 for each decreas- ing net (V

) of clopen subsets of X with T V

= ∅. In this case we write V

↓ ∅. We denote by M

(X) the space of all τ -additive members of M (X). Analogously, we denote by M

(X) the space of all σ-additive m, i.e. those m with m(V

) → 0 when V

↓ ∅. For an m ∈ M (X, E

) and s ∈ E, we denote by ms the element of M (X) defined by (ms)(V ) = m(V )s. A subset G of X is called a support set of an m ∈ M (X, E

) if m(V ) = 0 for each V ∈ K(X) disjoint from G.

Theorem 1 ([17). , Theorem 2.1] Let m ∈ M (X, E

) be such that ms ∈ M

(X), for all s ∈ E, and let p ∈ cs(E) with kmk

< ∞. Then :

(1) m

(V ) = sup

N

(x) for every V ∈ K(X).

(2) The set

supp(m) = \

{V ∈ K(X) : m

(V

) = 0}

is the smallest of all closed support sets for m.

(3) supp(m) = {x : N

(x) 6= 0}.

(4) If V is a clopen set contained in the union of a family (V

)

of clopen sets, then m

(V ) ≤ sup{m

(V

) : i ∈ I}.

Next we recall the definition of the integral of an f ∈ E

with respect to an m ∈ M (X, E

).

For a non-empty clopen subset A of X, let D

be the family of all α = {A

, A

, . . . , A

; x

, x

, . . . , x

}, where {A

, . . . , A

} is a clopen partition of A and x

∈ A

. We make D

into a directed set by defining α

≥ α

iff the partition of A in α

is a refinement of the one in α

. For an α = {A

, A

, . . . , A

; x

, x

, . . . , x

} ∈ D

and m ∈ M (X, E

), we define

ω

(f, m) = X

m(A

)f (x

).

If the limit lim ω

(f, m) exists in K, we will say that f is m-integrable over A and denote this limit by R

A

f dm. We define the integral over the empty set to be 0. For A = X, we write simply R f dm. It is easy to see that if f is m-integrable over X, then it is m-integrable over every clopen subset A of X and R

A

f dm = R

χ

f dm. If τ

is the topology of uniform convergence, then every m ∈ M (X, E

) defines a τ

-continuous linear functional φ

on C

(X, E), φ

(f ) = R f dm. Also every φ ∈ (C

(X, E), τ

)

is given in this way by some m ∈ M (X, E

).

For p ∈ cs(E), we denote by M

(X, E

) the space of all m ∈ M

(X, E

) for which m

is tight, i.e. for each ǫ > 0, there exists a compact subset Y of X such that m

(A) < ǫ if the clopen set A is disjoint from Y . Let

M

(X, E

) = [

p∈cs(E

M

(X, E

).

Every m ∈ M

(X, E

) defines a β

-continuous linear functional u

on C

(X, E), u

(f ) = R f dm. The map m 7→ u

, from M

(X, E

) to (C

(X, E), β

)

, is an algebraic isomorphism.

For m ∈ M

(X) and f ∈ K

, we will denote by (V R) R

f dm the integral of f , with respect to m, as it is defined in [25]. We will call (V R) R

f dm the (V R)-integral of f .

For all unexplained terms on locally convex spaces, we refer to [23] and [25].

2 Q-Integrals

We will recall next the definition of the Q-integral which was given in [14]. Let m ∈ M (X, E

) be such that ms ∈ M

(X) for all s ∈ E. This in particular happens if m ∈ M

(X, E

). For f ∈ E

and x ∈ X, we define

Q

(x) = inf

x∈V ∈K(X)

sup{|m(B)f (x)| : V ⊃ B ∈ K(X)}, kf k

= sup

x∈X

Q

(x).

Let S(X, E) be the linear subspace of E

spanned by the functions χ

s, s ∈ E, A ∈ K(X), where χ

is the K-characteristic function of A. We will write simply S(X) if E = K.

Lemma 1. If g ∈ S(X, E), then kgk

= sup

x∈X

Q

(x) < ∞.

Proof: The proof was given in [14], Lemma 7.2. Note that, if kmk

< ∞ and d ≥ kgk

, then Q

(x) ≤ d · m

(X).

Lemma 2. For g ∈ S(X, E), we have     Z

g dm  

  ≤ kgk

. Proof: Assume first that g = χ

s, A ∈ K(X). Then

|m(A)s| ≤ |ms|(A) = sup

y∈A

N

(y).

But, for y ∈ A, we have N

(y) = inf

y∈V ∈K(X)

sup

V ⊃B∈K(X)

|m(B)s| = inf

y∈V ∈K(X)

sup

V ⊃B∈K(X)

|m(B)g(y)| = Q

(y).

Thus |m(A)s| ≤ sup

Q

(y). In the general case, there are pairwise disjoint clopen sets A

, . . . , A

covering X and s

∈ E with g = P

k=1

χ

s

. Thus,  

  Z

g dm     =

X

m(A

)s

     ≤ max

1≤k≤n

|m(A

)s

| ≤ sup

x∈X

Q

(x) = kgk

.

Definition 1. Let m ∈ M (X, E

) be such that ms ∈ M

(X) for all s ∈ E. A function f ∈ E

is said to be Q-integrable with respect to m if there exists a sequence (g

) in S(X, E) such that kf − g

k

→ 0. In this case, the Q-integral of f is defined by

(Q) Z

f dm = lim

n→∞

Z g

dm.

If f is Q-integrable with respect to m, then for A ∈ K(X) the function χ

f is also Q-integrable. We define

(Q) Z

A

f dm = (Q) Z

χ

f dm.

As it is proved in [14], the Q-integral is well defined. If µ ∈ M

(X) and g ∈ K

, then Q

(x) = |g(x)|N

(x). Thus the Q-integral with respect to µ coincides with the integral as it is defined in [23], which we will call (VR)-integral. Hence

(V R) Z

g dµ = (Q) Z

g dµ.

Lemma 3. If f ∈ E

is Q-integrable with respect to an m ∈ M (X, E

) and if (g

) is a sequence in S(X, E), with kf − g

k

→ 0, then

kf k

= lim

n→∞

kg

k

< ∞, and    (Q)

Z f dm

  ≤ kf k

. Proof: Since

Q

(x) ≤ max{Q

(x), Q

(x)}, it follows that

kh + gk

≤ max{khk

, kgk

}.

Thus

kf k

≤ max{kf − g

k

, kg

k

} ≤ kf − g

k

+ kg

k

< ∞.

It follows that

|kf k

− kg

k

| ≤ kf − g

k

→ 0.

Moreover,    (Q) Z

f dm     = lim

    Z

g

dm  

  ≤ lim

kg

k

= kf k

. Hence the result follows.

Theorem 2. Let m ∈ M (X, E

) be such that ms ∈ M

(X) for all s ∈ E, and let f ∈ E

be Q-integrable. Define

m

: K(X) → K, m

(A) = (Q) Z

A

f dm.

Then m

∈ M

(X).

Proof: Since |m

(A)| ≤ kf k

, it is easy to see that m

∈ M (X). Let now V

↓ ∅ and ǫ > 0. Choose a g = P

k=1

χ

s

∈ S(X, E) such that kf − gk

< ǫ. Then Z

V_δ

g dm = X

(ms

)(V

∩ A

) → 0.

Let δ

be such that    R

Vδ

g dm    < ǫ if δ ≥ δ

. Now, for δ ≥ δ

, we have  

 (Q) Z

V_δ

f dm  

  ≤ max{

   (Q)

Z

V_δ

(f − g) dm     ,

    Z

V_δ

g dm    }

≤ max{kf − gk

,     Z

V_δ

g dm    } < ǫ.

Thus m

(V

) → 0.

Lemma 4. If f ∈ E

is Q-integrable with respect to an m ∈ M (X, E

), then the map x → Q

(x) is upper semicontinuous.

Proof: We need to show that, for each α > 0, the set V = {x : Q

(x) < α}

is open. So let x ∈ V and choose ǫ > 0 such that Q

(x) < α − 2ǫ. Let g ∈ S(X, E) be such that kf − gk

< ǫ. Let A

, . . . , A

be a clopen partition of X and s

∈ E such that g = P

k=1

χ

s

. Let k be such that x ∈ A

. There exists a clopen set B, containing x and

contained in A

, such that |m(D)g(x)| < Q

(x) + ǫ for every clopen set D contained in B.

If y ∈ B, then for B ⊃ D ∈ K(X) we have

|m(D)g(y)| = |m(D)g(x)| < Q

(x) + ǫ

≤ max{Q

(x), Q

(x)} + ǫ

≤ Q

(x) + 2ǫ.

Thus Q

(y) ≤ Q

(x) + 2ǫ < α. Hence x ∈ B ⊂ V and the result follows.

Lemma 5. If f ∈ E

is Q-integrable with respect to an m ∈ M (X, E

), then N

≤ Q

.

Proof: Let x ∈ X and ǫ > 0. In view of the preceding Lemma, there exists a clopen neighborhood V of X such that Q

(y) ≤ Q

(x) + ǫ for all y ∈ V . If V ⊃ B ∈ K(X), then

|m

(B)| ≤ sup

y∈B

Q

(y) ≤ Q

(x) + ǫ and so

N

(x) ≤ |m

|(V ) ≤ Q

(x) + ǫ.

Hence the result follows.

Lemma 6. Let m ∈ M (X, E

) be such that ms ∈ M

(X) for all s ∈ E. If g ∈ S(X, E), then Q

= N

.

Proof: Let {A

, . . . , A

} be a clopen partition of X and s

∈ E such that g = P

k=1

χ

s

. Suppose that N

(x) < α. Then, there exists a clopen neighborhood V of x such that

|m

|(V ) < α. Let x ∈ A

. If B is a clopen set contained in A

∩ V , then m

(B) = (Q)

Z

B

g dm = Z

B

g dm = m(B)g(x) since g = g(x) on B. Thus

Q

(x) ≤ sup

B⊂A_k∩V

|m(B)g(x)| ≤ |m

|(V ) < α.

This proves that Q

≤ N

and the result follows.

Theorem 3. If f ∈ E

is Q-integrable with respect to an m ∈ M (X, E

), then Q

= N

.

Proof: Assume that N

(x) < α and let 0 < ǫ < α. There exists a clopen neighborhood V of x such that |m

|(V ) < α. Let g ∈ S(X, E) be such that kf − gk

< ǫ. For A clopen contained in V , we have

|m

(A) − m

(A)| =    (Q)

Z

(f − g) dm  

  ≤ kf − gk

< ǫ and so

|m

(A)| ≤ max{ǫ, |m

(A)|} < α.

Thus

Q

(x) = N

(x) ≤ |m

|(V ) ≤ α.

Now

Q

(x) ≤ max{Q

(x), Q

(x)} ≤ α,

which proves that Q

≤ N

and the result follows by Lemma 5.

Theorem 4. Let m ∈ M (X, E

) be such that ms ∈ M

(X), for all s ∈ E, and let f ∈ E

be Q-integrable with respect to m. If g ∈ K

is Q-integrable with respect to m

, then gf is Q-integrable with respect to m and

(Q) Z

gf dm = (Q) Z

g dm

. Proof: If h ∈ K

, then

Q

(x) = |h(x)|Q

(x) = |h(x)|N

(x) = Q

(x).

Let (g

) be a sequence in S(X) such that kg − g

k

→ 0. We have kg − g

k

= sup

x∈X

|g(x) − g

(x)| · N

(x)

= sup

x∈X

Q

(x) = kgf − g

f k

. If A ∈ K(X), then χ

f is Q-integrable with respect to m and

(Q) Z

χ

f dm = (Q) Z

A

f dm = m

(A) = Z

χ

dm

. It follows that, for all n, g

f is Q-integrable with respect to m and

(Q) Z

g

f dm = Z

g

dm

→ (Q) Z

g dm

.

Since g

f is Q-integrable with respect to m and kgf − g

f k

→ 0, it follows that gf is Q-integrable and

(Q) Z

gf dm = lim

n→∞

(Q) Z

g

f dm = lim

n→∞

Z

g

dm

= (Q) Z

g dm

, which completes the proof.

Theorem 5. Let m ∈ M (X, E

) be such that ms ∈ M

(X), for all s ∈ E, and let p ∈ cs(E) with kmk

< ∞. If f ∈ E

is Q-integrable with respect to m, then, given ǫ > 0, there exists α > 0 such that 

(Q) R

A

f dm 

 < ǫ if m

(A) < α.

Proof: Let g ∈ S(X, E) with kf − gk

< ǫ. For a clopen set A, we have   R

A

g dm   ≤ kgk

· m

(A). Let α > 0 be such that α · kgk

< ǫ. If m

(A) < α, then

   (Q)

Z

A

f dm  

  ≤ max{

   (Q)

Z

A

(f − g) dm     ,

    Z

A

g dm    }

≤ max{kf − gk

, kgk

· m

(A)} < ǫ.

Lemma 7. Let m ∈ M

(X) and let g ∈ K

be (VR)-integrable. Then, given ǫ > 0, there exists δ > 0 such that kgk

≤ ǫ if |m|(A) < δ.

Proof: There exists h ∈ S(X) such that kg − hk

≤ ǫ. It suffices to choose δ > 0 such that δ · khk < ǫ.

Let m ∈ M (X). For A ⊂ X, we define

|m|

(A) = inf{|m|(V ) : V ∈ K(X), A ⊂ V }.

Recall that a sequence (g

) in K

converges in measure to an f ∈ K

, with respect to m (see [14], Definition 2.12) if, for each α > 0, we have

n→∞

lim |m|

{x : |g

(x) − g(x)| ≥ α} = 0.

Theorem 6 (Dominated Convergence Theorem). Let m ∈ M

(X) and let (f

) be a sequence of (VR)-integrable, with respect to m, functions, which converges in measure to some f ∈ K

. If there exists a (VR)-integrable function g ∈ K

such that |f

| ≤ |g| for all n, then f is (VR)-integrable and

(V R) Z

f dm = lim

n→∞

(V R) Z

f

dm.

Proof: Let ǫ > 0 and choose inductively n

< n

< . . . such that |m|

(V

) < 1/k, where V

= {x : |f

(x) − f (x) ≥ 1/k}.

Let V = T

S

k≥N

V

. If x ∈ V , then N

(x) = 0. Indeed, for every N , there exists k ≥ N with x ∈ V

and so N

(x) ≤ |m|(V

) < 1/k ≤ 1/N , which proves that N

(x) = 0. Also, for x ∈ X \ V , we have f (x) = lim

f

(x). In fact, there exists N such that x / ∈ V

for k ≥ N and so |f

(x) − f (x)| < 1/k → 0. It follows that |f (x)| ≤ |g(x)| when x / ∈ V . Since g is (VR)-integrable, there exists (by the preceding Lemma) δ > 0 such that kgk

< ǫ if

|m|(A) < δ. Let now α > 0 be such that α · kmk < ǫ. For each n, let G

= {x : |f

(x) − f (x)| ≥ α}

and choose a clopen set W

containing G

with |m|(W

) < 1/n+|m|

(G

). Since |m|

(G

) → 0, there exists n

such that |m|(W

) < δ if n ≥ n

. Let now n ≥ n

and x ∈ X. If x ∈ V , then N

(x) = 0. Suppose that x / ∈ V . Then |f (x)| ≤ |g(x)| and so

|f (x) − f

(x)|N

(x) ≤ |g(x)|N

(x).

If x ∈ W

, then |g(x)|N

(x) ≤ ǫ, since |m|(W

) < δ, while for x / ∈ W

we have

|f (x) − f

(x)|N

(x) ≤ α · kmk < ǫ.

Thus, for n ≥ n

, we have kf − f

k

≤ ǫ. Since f

is (VR)-integrable, it follows that f is (VR)-integrable and

(V R) Z

f dm = lim

n→∞

(V R) Z

f

dm

since    (V R)

Z

(f − f

) dm  

  ≤ kf − f

k

→ 0.

This completes the proof.

Let now τ be the topology of X and let K

(X) be the collection of all subsets A of X such that A ∩ Y is clopen in Y for each compact subset Y of X. It is easy to see that if A, A

, A

are in K

(X), then each of the sets A

, A

∩ A

and A

∪ A

is also in K

(X). Now K

(X) is a base for a zero-dimensional topology τ

on X finer than τ . We will denote by X

the set X equipped with the topology τ

. We have the following easily established

Theorem 7. (1) τ and τ

have the same compact sets.

(2) τ and τ

induce the same topology on each τ -compact subset of X.

(3) A subset B of X is τ

-clopen iff B ∈ K

(X).

(4) If Y is a zero-dimensional topological space and f : X → Y , then f is τ

-continuous iff the restriction of f to every compact subset of X is τ -continuous.

Let now m ∈ M (X, E

) be such that ms ∈ M

(X) for each s ∈ E.

Lemma 8. If B ∈ K

(X), s ∈ E and h = χ

s, then h is Q-integrable with respect to m.

Proof: Let ǫ > 0. Since ms ∈ M

(X), there exists a compact subset Y of X such that

|ms|(V ) < ǫ for each clopen subset V of X disjoint from Y . Since B ∩Y is clopen in Y and Y is compact, there exists A ∈ K(X) with B ∩ Y = A ∩ Y (see [25], p. 188). Let g = χ

s, f = h − g.

If x ∈ A ∆ B, then x is not in Y and so there exists V ∈ K(X) such that x ∈ V ⊂ Y

. If W ∈ K(X) is contained in V , then |m(W )f (x)| = |m(W )s| ≤ |ms|(V ) < ǫ and so Q

(x) ≤ ǫ.

Thus kh − gk

≤ ǫ. Hence the Lemma follows.

Now for B ∈ K

(X), we define

m

(B) : E → K, m

(B)s = (Q) Z

χ

s dm.

Clearly m

is linear. Let p ∈ cs(E) be such that m

(X) < ∞.

Theorem 8. Let A ∈ K

(X), and let V ∈ K(X) with A ⊂ V . Then : (1) |m

(A)s| ≤ |ms|(V ) ≤ m

(V ) · p(s) for all s ∈ E.

(2) m

∈ M

(X

, E

).

(3) m

s ∈ M

(X

) for all s ∈ E.

(4) If m ∈ M

(X, E

), then m

∈ M

(X

, E

).

Proof: Let s ∈ E, h = χ

s and x ∈ A ⊂ V . If W is a clopen subset of X contained in V , then |m(W )h(x)| ≤ |ms|(V ) and so Q

(x) ≤ |ms|(V ), which implies that

|m

(A)s| ≤ sup

x∈A

Q

(x) ≤ |ms|(V ) ≤ m

(V ) · p(s).

This proves that m

(A) ∈ E

and km

(A)k

≤ m

(V ). Clearly m

∈ M

(X

, E

) and km

k

≤ kmk

.

Let now s ∈ E and ǫ > 0. There exists a compact subset Y of X such that |ms|(Z) < ǫ for each Z ∈ K(X) disjoint from Y . Let B ∈ K

(X) be disjoint from Y and let x ∈ B. Then x / ∈ Y and so there exists a D ∈ K(X) containing x ad contained in Y

. For h = χ

s, we have Q

(x) ≤ |ms|(D) < ǫ. Thus |m

(A)s| ≤ ǫ. It follows that |m

s|(B) ≤ ǫ for each B ∈ K

(X) disjoint from Y and so m

s ∈ M

(X

). Finally, assume that m ∈ M

(X.E).

Given ǫ > 0, there exists a compact subset Y of X such that m

(V ) < ǫ for each V ∈ K(X) disjoint from Y . If s ∈ E, with p(s) > 0, then for V ∈ K(X) disjoint from Y we have |ms|(V ) ≤ m

(V ) · p(s) < ǫ · p(s). Thus, for B ∈ K

(X) disjoint from Y we have |m

s|(B) ≤ ǫ · p(s) and so m

(B) ≤ ǫ. This clearly completes the proof.

Theorem 9. Let m ∈ M (X, E

) be such that ms ∈ M

(X) for each s ∈ E. Then:

(1) If A ∈ K(X), then |ms|(A) = |m

s|(A) for all s ∈ E.

(2) If m ∈ M

(X, E

), then m

(A) = m

(A) for each A ∈ K(X).

(3) If f ∈ E

is Q-integrable with respect to m, then f is Q-integrable with respect to m

and Q

≤ Q

. Moreover

(Q) Z

f dm = (Q) Z

f dm

.

Proof: Let A ∈ K(X). Clearly |ms|(A) ≤ |m

s|(A). On the other hand, let |m

s|(A) >

θ > 0. There exists D ∈ K

(X), D ⊂ A, such that |m

(D)s| > θ. Let h = χ

s. Since

|m

(D)s| ≤ sup

Q

(x), there exists x ∈ D such that Q

(x) > θ. The set Y =

{z ∈ X : Q

(z) ≥ θ} is compact. Hence there exists Z ∈ K(X) with Z ∩ Y = D ∩ Y .

Since x ∈ Z ∩ A and Q

(X) > θ, there exists W ∈ K(X) contained in Z ∩ A and such

that |m(W )h(x)| > θ. Then h(x) = s and so |m(W )s| > θ, which proves that |ms|(A) > θ.

Thus, |ms|(A)| ≥ |m

s|(A). Assume next that m

(A) > α > 0. There exists B ∈ K

(X) contained in A and s ∈ E with |m

(B)s|/p(s) > α. Now |ms|(A) = |m

s|(A) > α · p(s).

Thus m

(A) ≥ |ms|(A)/p(s) > α, which shows that m

(A) = m

(A). Thus (1) and (2) hold.

(3). Assume that f ∈ E

is Q-integrable with respect to m.

Claim : If x ∈ D ∈ K(X), then sup

Z∈Kc(X),Z⊂D

|m

(Z)f (x)| = sup

Z∈K(X),Z⊂D

|m(Z)f (x)|.

Indeed, suppose that there exists a Z ∈ K

(X) contained in D such that |m

(Z)f (x)| > θ > 0.

For h = χ

f (x), we have

θ < |m

(Z)f (x)| ≤ sup

z∈Z

Q

(z).

Thus, there exists z ∈ Z with Q

(z) > θ. Since z ∈ Z ⊂ D, there exists W ∈ K(X) contained in D such that |m(W )h(z)| = |m(W )f (x)| > θ. This clearly proves the claim. Now

Q

(x) = inf

x∈D∈K(X)

sup

D⊃Z∈K(X)

|m(Z)f (x)|

= inf

x∈D∈K(X)

sup

D⊃Z∈Kc(X)

|m

(Z)f (x)| ≥ Q

(x).

Since f is Q-integrable with respect to m, there exists a sequence (g

) ⊂ S(X, E) ⊂ S(X

, E) such that kf −g

k

→ 0. But then kf −g

k

m(k)

≤ kf −g

k

→ 0. Hence f is Q-integrable with respect to m

and

(Q) Z

f dm

= lim

n→∞

Z

g

dm

= lim

n→∞

Z

g

dm = (Q) Z

f dm.

This completes the proof of the Theorem.

Next we recall the definition of the topology ¯ β

which was given in [14]. Let C

(X, E) be the space of all bounded E-valued functions on X whose restriction to every compact subset of X is continuous. By Theorem 7 we have that C

(X, E) = C

(X

, E). For p ∈ cs(E), we denote by ¯ β

the locally convex topology on C

(X, E) generated by the seminorms f 7→ khf k

, h ∈ B

(X). Since X and X

have the same compact sets, we have that B

(X) = B

(X

) and so ¯ β

coincides with the topology β

on C

(X

, E). The topology β ¯

is defined to be the locally convex projective limit of the topologies ¯ β

, p ∈ cs(E). Thus β ¯

coincides with topology β

on C

(X

, E).

Theorem 10. (1) If m ∈ M

(X, E

), then every f ∈ C

(X, E) is Q-integrable with respect to m and

(Q) Z

f dm = Z

f dm

. Thus the map

φ

: C

(X, E) → K, φ

(f ) = (Q) Z

f dm is ¯ β

-continuous.

(2) If E is polar, then every ¯ β

-continuous linear functional φ on C

(X, E) is of the form

φ

for some m ∈ M

(X, E

).

Proof: 1. Let p ∈ cs(E) be such that m ∈ M

(X, E

) and kmk

< 1. Let d > kf k

and ǫ > 0. There exists a compact subset Y of X such that m

(V ) < ǫ/d for every V ∈ K(X) disjoint from Y . For each x ∈ Y , the set

D

= {y ∈ Y : p(f (y) − f (x)) < ǫ}

is clopen in Y and D

= D

if D

∩ D

6= ∅. In view of the compactness of Y , there are x

, . . . , x

in Y such that the sets D

, . . . , D

form a partition of Y . For each k, there exists a clopen subset V

of X such that V

∩ Y = D

. If W

= V

\ S

i6=k

V

, then W

∩ Y = D

. Let g = P

k=1

χ

f (x

). Then kf − gk

≤ ǫ. Indeed, let x ∈ X.

Case I: x / ∈ Y . There is a clopen neighborhood V of x disjoint from Y . If B ∈ K(X) is contained in V , then

|m(B)[f (x) − g(x)]| ≤ p(f (x) − g(x)) · m

(V ) ≤ ǫ and so Q

(x) ≤ ǫ.

Case II : x ∈ Y . There exists a k such that x ∈ W

and so g(x) = f (x

). If a clopen set B is contained in W

, then

|m(B)[f (x) − g(x)]| = |m(B)[f (x) − f (x

]| ≤ m

(V

) · p(f (x) − f (x

)) ≤ ǫ, and so again Q

(x) ≤ ǫ. This proves that kf − gk

≤ ǫ and so f is Q-integrable. Now

φ

(f ) = (Q) Z

f dm = (Q) Z

f dm

= Z

f dm

. Thus φ

is ¯ β

-continuous on C

(X, E).

Finally assume that E is polar and let φ be a ¯ β

-continuous linear functional on C

(X, E).

Since ¯ β

induces the topology β

on C

(X, E), there exists an m ∈ M

(X, E

) such that φ(f ) =

Z

f dm = (Q) Z

f dm

for each f ∈ C

(X, E). Now φ and φ

are both ¯ β

-continuous on C

(X, E) and they coincide on the ¯ β

-dense subspace C

(X, E) of C

(X, E). Thus φ = φ

and the proof is complete.

3 The Dual Space of (C b (X, E), β 1 )

For u a linear functional on C

(X, E), p ∈ cs(E) and h ∈ K

, we define

|u|

(h) = sup{|u(g)| : g ∈ C

(X, E), p ◦ g ≤ |h|}.

Theorem 11. For a linear functional u on C

(X, E), the following are equivalent : (1) u is β

-continuous.

(2) For each sequence (V

) of clopen sets, with V

↓ ∅, there exists p ∈ cs(E) such that kuk

< ∞ and lim

|u|

(χ

) = 0.

(3) For each sequence (h

) in C

(X), with h

↓ 0, there exists p ∈ cs(E) such that kuk

< ∞ and lim

|u|

(h

) → 0.

Proof: (1) ⇒ (2). Let V

↓ ∅ and H = T V

. Then H ∈ Ω

and so u is β

-continuous for some p ∈ cs(E). Let ǫ > 0 and h ∈ C

be such that

W

= {f ∈ C

(X, E) : khf k

≤ 1} ⊂ W = {f : |u(f )| ≤ ǫ}.

It is easy to see that kuk

< ∞. Let M = {x ∈ X : |h(x)| ≥ 1}. There exists n

such that M ⊂ V

. Let now n ≥ n

and f ∈ C

(X, E) with p ◦ f ≤ |χ

|. Let f

= χ

f, f

= f − f

. If x ∈ M , then x ∈ V

and so p(f (x)) = 0. This implies that f

∈ W

⊂ W . Also, if x / ∈ M , then |h(x)| ≤ 1 and so |h(x)|p(f (x)) ≤ 1, which proves that f

∈ W

. Thus f = f

+ f

∈ W , which shows that |u|

(χ

) ≤ ǫ.

(2) ⇒ (3). Let h

↓ 0. Without loss of generality, we may assume that kh

k ≤ 1. Let λ ∈ K, 0 < |λ| < 1 and set

V

= {x : |h

(x) ≥ |λ|}.

Then V

↓ ∅. By (2), there exists p ∈ cs(E) with kuk

< ∞ and |u|

(χ

) → 0. We may choose p so that kuk

≤ 1. Choose n

such that |u|

(χ

) < |λ| if n ≥ n

. Let now n ≥ n

. We will show that |u|

(h

) ≤ |λ|. In fact, let f ∈ C

(X, E) with p ◦ f ≤ |h

|, g

= χ

f , g

= f − g

. If x ∈ V

, then p(g

(x)) ≤ |h

(x)| and so p ◦ g

≤ |χ

|, which implies that |u(g

)| ≤ |λ|.

If x / ∈ V

, then p(g

(x)) = p(f (x)) ≤ |h

(x)| < |λ|. Hence |u(g

)| ≤ kuk

· kg

k

≤ |λ|, and therefore |u(f )| ≤ |λ|. This proves that |u|

(h

) ≤ |λ|.

(3) ⇒ (2). It is trivial.

(2) ⇒ (1). Let

W = {f ∈ C

(X, E) : |u(f )| ≤ 1}

and let H ∈ Ω

. There exists a decreasing sequence (V

) of clopen subsets of X with T V

= H. Let p ∈ cs(E) be such that kuk

≤ 1 and |u|

(χ

) → 0. Let λ be a nonzero element of K and choose n so that |u|(χ

) < |λ|

. Now

W

= {f ∈ C

(X, E) : kf k

≤ |λ|, kf k

≤ 1} ⊂ W.

Indeed, let f ∈ W

and set f

= χ

f, f

= f − f

. Since |λ

f

| ≤ |χ

|, we have that

|u(f

)| ≤ 1. Also |u(f

)| ≤ kf

k

≤ 1, and so |u(f )| ≤ 1, which proves that W

⊂ W . By [13], Theorem 2.2, it follows that W is a β

-neighborhood of zero. This, being true for all H ∈ Ω

, implies that W is a β

-neighborhood of zero, i.e. u is β

-continuous, which completes the proof.

Theorem 12. For a set H of linear functionals on C

(X, E), the following are equivalent :

(1) H is β

-equicontinuous.

(2) If (V

) is a sequence of clopen subsets of X which decreases to the empty set, then there exists p ∈ cs(E) such that sup

kuk

< ∞ and |u|

(χ

) → 0 uniformly for u ∈ H.

(3) If (h

) is a sequence in C

(X) with h

↓ 0, then there exists p ∈ cs(E) such that sup

kuk

< ∞ and |u|

(h

) → 0 uniformly for u ∈ H.

Proof: (1) ⇒ (2). Let V

↓ ∅. Then Z = T V

∈ Ω

. Let λ ∈ K, λ 6= 0. Since H is β

-equicontinuous, the set λH

is a β

-neighborhood of zero. Thus, there exists p ∈ cs(E) such that λH

is a β

-neighborhood of zero. Let h ∈ C

be such that

W

= {f : khf k

≤ 1} ⊂ λH

.

It follows now easily that sup

kuk

< ∞. Also, as in the proof of the implication (1) ⇒ (2) in the preceding Theorem, we prove that |u|

(χ

) → 0 uniformly for u ∈ H. For the proofs of the implications (2) ⇒ (3) ⇒ (2) ⇒ (1) we use an argument analogous to the one used in the proof of the preceding Theorem.

Theorem 13. In the space C

(X), β

is the finest of all locally solid topologies γ with the following property: If (f

) ⊂ C

(X) with f

↓ 0, then f

→ 0.

Proof: By [12], Theorems 3.7 and 3.8, β

is locally solid and f

→ 0 when f

↓ 0. Consider now the family U of all solid absolutely convex subsets W of C

(X) such that f

∈ W eventually when f

↓ 0. Clearly U is a base at zero for the finest locally solid topology γ

on C

(X) having the property mentioned in the Theorem.

Claim I : γ

is coarser than τ

. Indeed, let W ∈ U and let λ ∈ K, 0 < |λ| < 1. For each n, let g

be the constant function λ

. Since g

↓ 0, there exists an n with g

∈ W . If now f ∈ C

(X) with kf k ≤ |λ|

, then f ∈ W , which implies that W is a τ

-neighborhood of zero.

Claim II : β

is finer than γ

and hence β

= γ

. Indeed, let W ∈ U, Z ∈ Ω

and r > 0.

There exists ǫ > 0 such that

W

= {f ∈ C

(X) : kgk ≤ ǫ} ⊂ W.

Choose µ ∈ K with |µ| ≥ r. There exists a decreasing sequence (V

) of clopen subsets of X with Z = T V

βoX

. Since µχ

↓ 0, there exists n such that µχ

∈ W . Let now f ∈ C

(X) with kf k ≤ r, kf k

≤ ǫ, and let g = f · χ

, h = f − g. Then |g| ≤ |µχ

|and so g ∈ W since W is solid. Also, khk ≤ ǫ and so h ∈ W , which implies that f ∈ W . This proves that W is a β

-neighborhood of zero for all Z ∈ Ω

and hence W is a β

-neighborhood of zero. This clearly completes the proof.

The proofs of the following two Theorems are analogous to the ones of Theorems 12 and 13.

Theorem 14. For a subset H of linear functionals on C

(X, E), the following are equiv- alent :

(1) H is β-equicontinuous.

(2) For each net (V

), of clopen subsets of X with V

↓ 0, there exists p ∈ cs(E) such that sup

kuk)p < ∞ and |u|

(χ

) → 0 uniformly for u ∈ H.

(3) For each net (h

) in C

(X) with h

↓ 0, there exists p ∈ cs(E) such that sup

kuk

<

∞ and |u|

(h

) → 0 uniformly for u ∈ H.

Theorem 15. In the space C

(X), β is the finest of all locally solid topologies γ with the following property: If (f

) ⊂ C

(X) with f

↓ 0, then f

→ 0.

4 The Space M b (X, E ^′ )

A subset A of X is called bounding if every f ∈ C(X) is bounded on A. Note that several authors use the term bounded set instead of bounding. But in this paper we will use the term bounding to distinguish from the notion of a bounded set in a topological vector space. A set A ⊂ X is bounding iff A

is compact. In this case (as it is shown in [1], Theorem 4.6) we have that A

= A

. Clearly a continuous image of a bounding set is bounding.

Theorem 16 ([17). , Theorem 3.4] If G is a locally convex space (not necessarily Haus- dorff ), then every bounding subset A of G is totally bounded.

We denote by M

(X, E

) the space of all m ∈ M (X, E

) which have a bounding support, i.e. there exists a bounding subset B of X such that m(V ) = 0 for all clopen V disjoint from B. In case E = K, we write simply M

(X).

Theorem 17. If m ∈ M

(X, E

), then every f ∈ C(X, E) is m-integrable. Moreover, if B is a bounding support of m and p ∈ cs(E) with m

(X) < ∞, then

    Z

f dm  

  ≤ kf k

· kmk

.

Proof: Let f ∈ C

(X, E) and let B be a bounding subset of X which is a support set for m. Since the closure of a bounding set is bounding, we may assume that B is closed. Let p ∈ cs(E) with m

(X) < ∞. The set f (B) is bounding in E and hence totaly bounded by Theorem 4.1. Thus, given ǫ > 0, there are x

, . . . , x

in B such that the sets

V

= {x : p(f (x) − f (x

)) ≤ ǫ/kmk

}, k = 1, . . . , n, are pairwise disjoint and cover B. Let V

= X \ S

k=1

V

and choose x

∈ V

if V

6= ∅.

Let {W

, . . . , W

} be a clopen partition of X which is a refinement of {V

, . . . , V

} and y

∈ W

. We may assume that S

i=1

V

= S

j=1

W

. If W

⊂ V

for some i ≤ n, then

|m(W

)[f (y

) − f (x

)]| ≤ kmk

· p(f (y

) − f (x

)) ≤ ǫ, while, for W

⊂ V

, we have m(W

) = 0. Thus

X

m(W

)f (y

) − X

m(V

)f (x

)      ≤ ǫ.

This proves that f is m-integrable and      Z

f dm − X

m(V

)f (x

)      ≤ ǫ.

Since |m(V

)f (x

)| ≤ kf k

· kmk

, it follows that  

  Z

f dm  

  ≤ max{kf k

· kmk

, ǫ},

for each ǫ > 0, and the proof is complete.

We denote by τ

the topology on C(X, E) of uniform convergence on the bounding subsets of X.

Lemma 9. The space S(X, E) is τ

-dense in C(X, E).

Proof: Let f ∈ C(X, E), p ∈ cs(E), ǫ > 0 and B a bounding subset of X. There are x

, . . . , x

in B such that the sets

V

= {x : p(f (x) − f (x

)) ≤ ǫ}, k = 1, . . . , n, are pairwise disjoint and cover B. If g = P

k=1

χ

f (x

), then kf − gk

≤ ǫ and the Lemma follows.

Theorem 18. For m ∈ M

(X, E

), let

ψ

: C(X, E) → K, ψ

(f ) = Z

f dm.

Then ψ

is τ

-continuous and M

(X, E

) is algebraically isomorphic to the dual space of (C(X, E), τ

) via the isomorphism m 7→ ψ

.

Proof: In view of Theorem 4.2, ψ

is an element of G = (C(X, E), τ

)

. On the other hand, let ψ ∈ G. Since τ

|

is coarser than the topology τ

of uniform convergence, there exists m ∈ M (X, E

) such that ψ(f ) = R

f dm for all f ∈ C

(X, E). Let B a bounding subset of X and p ∈ cs(E) be such that

{f ∈ C(X, E) : kf k

≤ 1} ⊂ {f : |ψ(f )| ≤ 1}.

It follows that B is a support set for m and so m ∈ M

(X, E

). Now ψ and ψ

are both τ

-continuous and they coincide on the τ

-dense subspace S(X, E) of C(X, E). Thus ψ = ψ

and the result follows.

Recall that, for p ∈ cs(E), M

(X, E

) denotes the space of all m ∈ M

(X, E

) such that m

(A

) → 0 for each decreasing net (A

) of clopen subsets of X for which T A

βoX

∈ Ω

(see [13], p. 123).

Theorem 19. Let m ∈ M

(X, E

). If p ∈ cs(E) is such that kmk

< ∞, then m ∈ M

(X, E

).

Proof: Let B be a bounding support for m and let (V

)

be a clopen partition of X.

The set B

is compact and

B

⊂ θ

X ⊂ [

i

V

.

Hence, there exists a finite subset J of I such that B

⊂ [

i∈J

V

and so B ⊂ S

i∈J

V

, which implies that m

( S

i /∈J

V

) = 0. Thus m ∈ M

(X, E

) by [13], Theorem 5.7.

Theorem 20. The topology induced by τ

on C

(X, E) is coarser than β

.

Proof: Let B be a bounding subset of X, p ∈ cs(E) and H ∈ Ω

. There exists a clopen partition (V

)

) of X such that

H ⊂ β

X \ [

i∈I

V

.

As in the proof of the preceding Theorem, there exists a finite subset J of I such that B ⊂ S

i∈J

V

= V . If h = χ

, then h

= χ

vanishes on H and {f ∈ C

(X, E) : khf k

≤ ǫ} ⊂ {f : kf k

≤ ǫ}

which clearly completes the proof.

5 M _s (X) as a Completion

The space M

(X) was introduced in [12]. It is the space of the so called separable members of M

(X). For m ∈ M (X), d a continuous ultrapseudometric on X and A a d-clopen subset of X, we define

|m|

(A) = sup{|m(B)| : B ⊂ A, B d − clopen}.

For F ⊂ X, we define

|m|

(F ) = inf sup

n

|m|

(A

),

where the infimum is taken over the family of all sequences (A

) of d-clopen sets which cover

F . An element m of M

(X) is said to be separable if, for each continuous ultrapseudometric

d on X, there exists a d-closed, d-separable subset G of X such that |m|

(X \ G) = 0. As it is

shown in [12], if m ∈ M

(X), then every f ∈ C

(X) is m-integrable. Let now G = (C

(X), τ

)

,

where τ

is the topology of uniform convergence. For each x ∈ X, let δ

be the corresponding

Dirac measure. Thus δ

∈ G, δ

(f ) = f (x). Let L(X) be the subspace of G spanned by the