General Vector Measure Theory

Introduction

It seems to be well forgotten that many of the first ideas in Geometry, the basis theory and isomorphic theory of Banach spaces have vector measure theoretic origins . Equally we forgotten is the fact that much of the early interest in the weak and weak* compactness was motivated by vector measure theoretic considerations. Vector measure help mainly to study uniform convex norm of absolute continuous function on Banach space. We present few main highlights of important work of Vector measure and describe in details about properties of Vector measure in abstract ways.

In 1936 , J A Clackson introduced the notion of uniform convexity to prove that absolute continuous function on a Euclidean space with values in a uniformly convex Banach space are the integral of their derivatives.

Get quality help now

KarrieWrites

Verified writer

Proficient in: Math

5 (339)

“ KarrieWrites did such a phenomenal job on this assignment! He completed it prior to its deadline and was thorough and informative. ”

+84 relevant experts are online

Hire writer

At the same time, clackson used vector measure-theoretic ideas to prove that many familiar Banach spaces donot admits equivalent uniformly convex norms. N .Dunford and A.P.Morse in 1936 introduced the notion of a boundedly complete basis to prove that absolutely continuous function on a Euclidean space with values in a Banach space with a boundedly complete basis are the integrals of their derivatives.

Shortly thereafter Dunford was able to recognize the Dunford Morse theorem and the Clackson theorem as genuine Radon –Nikodym theorem for Bochner integral. This was the first Radon –Nikodym theorem for vector measures on abstract measure spaces. B.J.Pettis in 1938 ,made his contribution to the Orlicz-pettis theorem for the purpose of proving that weakly countably additive vector measure are norm countably additive.

In 1938 I Gelford used vector measure theoretic methods to prove that 𝐿1[0,1]is not isomorphic to a dual of a Banach space.

In 1939, Pettis shows that the notions of weak and weak* compactness are intimately related to the problem of differentiating vector valued functions on Euclidean space. In 1940 Dunford and Pettis built on their earlier work to represent weakly compact operators on 𝐿1 to a separable dual space by means of Bochner integral. By means of their integral representation they were able to prove that 𝐿1 has the property now known as the Dunford –pettis property After the world war the work in vector measure theory become little more than formal generalizations of the scalar theory.

Representation theories for operators on function space become the vogue, but all too often the representation theories gave no new information about the operators they represented. During mid fifties A. Grothendieck used that ignoned vector measure theory of the late thirties and early forties to launch a monumental study of linear operators. The repercussions of Grothendieck’s work are still being felt today.

Also in the mid fifties Grothendieck , R.G. Bertle, N Dunford and J.T.Schwartz studied operators on spaces of continuous functions and proved the first important theorems in the theory of vector measure after 15 years. In early sixties , largely through the pioneering work of A. Pelezynski and J.Lindenstrauss , Banach space theory came back to life and today has re emerged as a deep and vigorous area of mathematical inquiry.

I n the mid sixties N- Dinculeanu gave an intensive study of many of the theorem of vector measure theory that had been proven between 1950 and 1965. Dinculeanu’s monography was the catalytic agent that the theory of vector measure needed. In 1974 J.Diestel and B. Faires give a new direction of vector measure. They presents four sections treat four different but somewhat relatedtopics in the theory of vector measures. First section is concerned with the theory of strongly bounded vector measure, the main result is to provides criteria for Banach space Xto possess the property that every X valued bounded additive map with values in X be strongly bounded base by Pettis theorem. Second section is concerned with the Jordan Decomposition of vector measures with the values in a Banach lattice. Third section deals with the Integrability of certain scalar functions with respect to a vector measure. Utilizing the series representation of a scalar function and its integral , a result of D.R Lewis is generalized –Also , a criterion for Integrability In 1975 A Katsaras defines spaces of vector measures.

Also during same time Igor Kluvanek work on the range of vector measure. In 1976 B.Faires and T.J.Morrison define some sort of Boolean algebra with vector values in vector lattice.In 1977 Hannu Niemi tried to work in Hilbert space anout compactness. In 1983 R.Rao and A.S.Sastry did work on product vector measure via Bartle Integrals.In 1985 L Drewnowski of Poznan investigate on almost basically scattered vector measures.

Horst osswald and yeneng sun work on two way of constructing countably additive vector measures from internal vector measures. The connection of the extendability of vector valued leob measures and the existence of the Internal control measure is shown in 1991

Vector measure :

Definition1: Let S be a set. A nonvoid class R of subsets of S is called a semi tribe (𝜎 𝑟𝑖𝑛𝑔) if (i) 𝐴 ,𝐵 ∈ 𝑅 → 𝐴∪ 𝐵𝜖𝑅 ,𝐴− 𝐵𝜖𝑅 (ii)𝐴𝑛 ∈ 𝑅(𝑛 = 1,2,… . ) → 𝐴𝑛 ∞ 𝑛=1 ∈ 𝑅 From this definitions it follows that a semi tribe R has the following properties (i) 𝐴𝑛 ∈ 𝑅 ,𝐴 ∈ 𝑅 → 𝐴𝑛 ⊂ 𝐴 𝑖 = 1,2,… . . 𝑡𝑕𝑒𝑛 𝐴𝑛 ∞ 𝑛=1 ∈ 𝑅 (ii) If we set 𝑅𝐴 = 𝐵 ∩ 𝐴 ∶ 𝐵 ∈ 𝑅 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑓𝑎𝑚𝑖𝑙𝑦 𝐴 ∈ 𝑅 then 𝑅𝐴 is a tribe on A.

Definition 2:

Suppose that X be a normed space and 𝑋 its completion. Let R be a clan( ring). A set function m defined on R with values in X is called a vector measure if the following conditions satisfied (i) 𝑚 ∅ = 0 (ii) For every sequence 𝐸𝑛 of mutually disjoint sets of R such that 𝐸 = 𝐸𝑛 ∞ 𝑛=1 ∈ 𝑅 Then 𝑚 𝐸 = 𝑚(𝐸𝑛) ∞ 𝑛=1 Example 1 of a finitely additive vector measure Let 𝑇:𝐿∞ [0,1] → 𝑋 be a continous linear operator. For each Lebesgue measurable set 𝐸 ⊑ 0,1 ,𝑑𝑒𝑓𝑖𝑛𝑒 𝐹[𝐸] to be T(𝛾𝐸) Then by linearlity of T , F is seen to be finitely additive vector measure which may even in the case that X is real numbers-fails to be countably additive Implies 𝐹 𝐸 = 𝑇 𝛾𝐸 Implies 𝐹(𝐸) ≤ 𝛾(𝐸) 𝑇 Implies 𝐹( 𝐸𝑛 − 𝐹(𝐸) ∞ 𝑛=1 ∞ 𝑛=1 ≤ 0. The simplest such general example of non countably additive measure is provided by considering any Hahn Banach extension to 𝐿∞ [0,1] of a point mass functional on C[0,1]

Definition3 :

Let 𝑚:𝐹 → 𝑋 be a vector measure. The variation of m is the extended non negative function 𝑚 whose values on a set E ∈ 𝐹 is given by 𝑚 𝐸 = 𝑠𝑢𝑝𝜋 𝑚(𝐸) 𝐴𝜖𝜋 where the supremum is taken over all partitions 𝜋 of E into a finite number of pairwise disjoint members of 𝐹 .If 𝑚 (Ω) < ∞ then 𝑚 is called a measure of bounded variation. The semi variation of m is the extended non negative function 𝑚 whose value on a set 𝐸 ∈ 𝐹 is given by 𝑚 𝐸 = sup⁡{ 𝑥∗𝑚 𝐸 :𝑥∗ ∈ 𝑋∗, 𝑥∗ ≤ 1} where 𝑥∗𝑚 is the variation of the real valued measure𝑥∗𝑚 . If 𝑚 (Ω) < ∞ then m will called a measure of bounded semivariation

Remark: (i) Variation of m is a monotone finitely additive function of 𝐹 (ii) Semi variation of 𝐹 is a monotone sub additive function on 𝐹 (iii) 𝑚 (𝐸) ≤ 𝑚 (𝐸) Example 1 is of measure of bounded variation

Ex2: A measure of bounded semi variation but not of bounded variation Let 𝑏𝑒 the 𝜎 field of Lebesgue measurable subsets of [0,1] and defined 𝑚: → 𝐿∞ 0,1 by m(E)=𝜒𝐸

Definition4 :

Let F be a field of subsets of Ω and 𝑚:𝐹 → 𝑋 be a vector measure and 𝜇 be a finite non negative real valued measure on F . If lim𝜇(𝐸)→0 𝑚 𝐸 = 0 then m is called 𝜇 continuous and denoted be m≪ 𝜇

Note: when 𝑚 ≪ 𝜇, some time 𝜇 is called a control measure for m or m is continuous with respect to 𝜇 or m is absolute continuous with respect to 𝜇.

Pettis Theorem:

Let 𝑏𝑒 a 𝜎 field, 𝑚: Σ → 𝑋 be a countably additive vector measure and 𝜇 be a finite non negative real valued measure on Σ ,then m is a 𝜇 continuos i.e lim𝜇(𝐸)→0 𝑚 𝐸 = 0 if and only if m vanishes on sets of 𝜇 measure zero. Important note is that this theorem fails for countably additive measures on fields.

Theorem1 :

Let {𝐹𝜏 :Σ → 𝑋 ∶ 𝜏𝜖𝑇} be a uniformly bounded family of countably additive vector measures on 𝜎 −field Σ . The family {𝐹𝜏: 𝜏𝜖𝑇} is uniformly countably additive is equal to uniformly strongly additive if only if there exists a non negative real valued countably additive measure 𝜇 on Σ such that {𝐹𝜏: 𝜏𝜖𝑇} is uniformly 𝜇 continous .that is lim𝜇(𝐸)→0 𝐹𝜏(𝐸) = 0 is uniformly in 𝜏𝜖𝑇

Bartle Dunford Schwartz :

Let F be a countably additive vector measure defined on a 𝜎 field Σ then there exists a non negative real valued countably additive measure 𝜇 𝑜𝑛 Σ such that 𝜇(𝐸) → 0 if and only if 𝐹 (𝐸) → 0 in the fact 𝜇 can be choosen so that 0≤ 𝜇 𝐸 ≤ 𝐹 𝐸 for all E 𝜖Σ Following immediately is a key structure theorem for countably additive vector measure on 𝜎 fields

Bartle Dunford Schwartz : Let F be a countably additive vector measure defined on a 𝜎 field Σ .then F has a relatively weakly compact range

The NikodymBoundedness theorem : Let Σ be a field of subsets of Ω and let {𝐹𝜏∶𝜏𝜖𝑇} be a family of X valued bounded vector measures defined on Σ .If 𝑠𝑢𝑝𝜏𝐹𝜏(𝐸) 0 ∃ a subsequence (𝐸𝑛𝑗 ) of (𝐸𝑛) such that 𝜇𝑛𝑗 ( 𝐸𝑛𝑘 ) 0 such that for 𝐴∈ Σ𝑤𝑖𝑡𝑕𝜇(𝐴) ≤𝜕 for all f 𝜖Γ∫ f dμ≤∈A

If Γ is a bounded subset of 𝐿1(𝜇) then Γ is equi-integrable is equivalent to

lim𝑐→∞𝑠𝑢𝑝Γ∈F ∫ΓΓ >𝑐𝑑𝜇 = 0

Lemma: Let Σ be a 𝜎−field of subsets of S and 𝜇 be a finite nonnegative countably additive measure on Σ . Any one of the following statements about a bounded linear operator 𝑇: (𝐿∞(𝜇) →𝑋) implies all others

• a) T is weak* continuous
• b) The representing measure of T is countably additive
• c) The representing measure of T is 𝜇−continuous

BartleDunford Schwartz : Let 𝑇:𝐶(Ω) →𝑋 be a bounded linear operator with representing measure G .Any one of the following statements all the others

• a) The operators T is weakly compact
• b) The measure G takes all its values in X
• c) The measure G is countably additive
• d) The measure G is strongly additive.

An Important Aspect come from this in probabilistic space by Dunford-pettis theorem as If (𝑋,Σ,𝜇) is probability space and F be a bounded subset of 𝐿1(𝜇) . F is equi-integrable if only if F is relatively compact subset of 𝐿1(𝜇) with the weak topology .

References

1.  On vector measure :Sachioohba(1970)
2.  On orthogonally scattered dilations of bounded vector measures : HannuNeimi
3. A Dunford-pettis theorem for 𝐿1/𝐻∞ : I. Crop & Delbaen (1976)
4.  Vector measure : J .Diestel & J.J Uhl.jr
5. Liapounoff’s vector measure theorem in Banach spaces and applications to general equilibrium theory : Michel Greinecker & Konrad Podczeck