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**Chapter 2 ****Liapounoff Convexity-type ****Theorems**

A. Liapounoff [Lia40] showed that if m is a σ-additive measure defined on a σ-field, Σ, taking values in a finite dimensional vector space, E1, we say m ∈ ca(Σ, E1), then the range of m denoted by Rm is compact and if m is non-atomic then Rm is convex, see also J. Lindenstrauss [Lin66]. Various well-known related theorems for infinite dimensional vector spaces exists. Some of these theorems are listed below. Let E be a quasi-complete locally convex space and m ∈ ca(Σ, E) a non-atomic measure:

• (I. Kluv´anek [Klu73a, Theorem 1, Corollary 3.1]. The weak closure of Rm coincides with the closure of co(Rm).

Let F be a Fr´echet space and m ∈ ca(Σ, F) a non-atomic measure:

• (S. Ohba [Ohb78] see also I. Kluv´anek and G. Knowles [KK76, Theorem IV.6.1] and [SS03]). If Rm is relatively compact then the closure of Rm is convex.

• (J.J. Uhl [Uhl69] generalized by S. Ohba [Ohb78]). If F has the Radon-Nikod´ym prop17 erty and if m is also of bounded variation then the closure of Rm is compact and convex.

In this chapter, these theorems are investigated for the case of finitely additive, bounded finitely additive and strongly additive vector measures defined on fields (of sets) and fields of sets with the interpolation property (I) of G.L Seever [See68]. In Section 2.1, we show that none of the above mentioned theorems may hold if the σ-field is replaced by a field. Here a property stronger than non-atomicity must be considered. A. Sobczyk and P.C. Hammer [SH44] utilized the concept of “continuous” set function.

To avoid confusion, we call this concept strongly continuous as done in [BRBR83]. In Section 2.2, we investigate the relationship between non-atomicity, strong continuity and Darboux properties for the case of non-negative finite measures defined on a fields of sets and fields of sets with property (I). The strong continuity property is introduced for the case of Fr´echet space-valued measures in Section 2.3. Finally in Section 2.4 we give the mentioned Liapounoff theorems and discuss conditions under which bounded finitely additive measures are strongly

additive.

**Counterexample**

The following example is la raison d’ˆetre for the structures studied in this chapter. This example shows that the classical Liapounoff Convexity theorem and the mentioned theorems by I. Kluv´anek , J.J. Uhl and S. Ohba can’t be extended to the case of a non-atomic vector measure on a field. In fact these theorems can’t even be extended to a non-atomic σ-additive vector measure of bounded variation on a field. Let Ω = [0, 1] and let F be the field generated by all sets of the form [a, b) where a < b and are rational numbers in Ω. Let α be any number in Ω. It is important to note that, since a σ-field isn’t under consideration, {α} ∈/ F. Let µ be the ”indicator” measure on F for the point α i.e. for any set A ∈ F if α ∈ A then µ(A) = 1 otherwise µ(A) = 0. Clearly, µ is an atomic measure. Let λ be the restriction of the Lebesgue measure on Ω to F. The non-negative measure λ is non-atomic, since for every set A ∈ F such that λ(A) > 0 there exists a subset B of A in F such that 0 < λ(B) < λ(A). The vector measure m :

is σ-additive since λ and µ are both σ-additive measures on F. For any π ∈ P(Ω, F), only one set in π, say set A, can contain the point α. Under the sup-norm of R2XD∈πkm(D)k∞ = km(A)k∞ +X D∈π,D6=Akm(D)k∞ < 2 Hence, the measure m is of bounded variation and thus also strongly additive, see [DU77, Proposition I.1.9]. Now, since λ is non-atomic, m is also non-atomic. It is obvious that Rm is neither compact, nor convex. Since the rational numbers are dense in the real numbers the closure of Rm denoted by Rgm is compact but non-convex. Let ˜m denote the extension of m to σ(F). Since Rm is dense in Rm˜ , it’s worth studying the relationship between m and ˜m, specifically the non-atomicity relationship. Although m on F is non-atomic, ˜m on σ(F) is atomic, since but {α} does not contain any non-empty subset. We call {α} an imbedded atom of F in terms of m. That is, an imbedded atom of a field F in terms of a vector measure m is a set in σ(F) which is an atom of ˜m, the extension of m to σ(F).

**Introduction 1**

**1 Preliminaries **

1.1 Locally Convex Spaces

1.1.1 Quotient and Normed Spaces

1.2 Vector Measures

1.2.1 Spaces of Measures

1.2.2 Stone Representation

1.2.3 p-semivariation

1.2.4 Bartle-Dunford-Schwartz-type Theorems

1.3 Polish Spaces

1.4 Vector-valued Measurable Functions

1.4.1 Integrability and Integrals

1.5 Nuclear maps and Nuclear spaces

**2 Liapounoff Convexity-type Theorems **

2.1 Counterexample

2.2 Non-negative Scalar Measures

2.3 Vector measures

2.4 Liapounoff Convexity-type Theorems

**3 Barrelled spaces **

3.1 Existence of the Dunford Integral

**4 Nuclear spaces and Nuclear maps **

4.1 Measures

4.2 Measurability and Integrability

**5 Factorization of Measurable Functions **

5.1 Core Results

5.2 Applications

5.2.1 Set-valued Operators

5.2.2 Operator-valued Measurable Functions

5.2.3 Conditional Expectation

5.2.4 Operators on L1(µ) and L1(µ, X)

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Measures and functions in locally convex spaces