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Integration in Lattice Spaces

Chapter One

PREAMBLE OF THE STUDY

Riemann-Stieltjes Integration

Definition of the Riemann-Stieltjes integral on a compact set

Consider an arbitrary function f : [a, b] → R.

The Riemann-Stieltjes integral of f on [a, b] associated with F , if it exists, is denoted by:

I = f (x) dF (x)

In establishing the existence of the Riemann-Stieltjes integral of a func- tion, we need the function to be bounded.

Next, we define the Riemann-Stieltjes sums. To do so, for each n ≥ 1, we divide [a, b] into l(n) sub-intervals (l ≥ 1).

Let π_n be a subdivision of [a, b] that divides[a, b] into l(n) sub-intervals. So,

]a, b] =

l(n)−1

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]xi,n, xi+1,n],

i=0

where a = x₀_,n < x₁_,n < … < x_l₍_n₎_,n = b.

CHAPTER TWO

Integration with respect to a measure on R : a summary

In this part, Considering a measure space (Ω, A, m) we are concerned with recalling the steps of the construction of the integral of a real-valued measurable function f : (Ω, A) → R¯

with respect to a measure, denoted by:

∫ f dm = ∫

f (ω) dm(ω) =

Ω

f (ω) m(dω)

Along the document, by ”The Real-valued Mapping Modern Integrals (RVM- MI)”, we mean the integrals of real-valued measurable functions.

The construction

STEP 1M: Definition of the integral for a non-negative elementary funtion f .

Let,

f =

i=1

α_i1_Ai (p ≥ 1, α_i ∈

R₊, A_i ∈ A, A₁ + A₂ + … + A_p = Ω) be a non-negative elementary function.

The integral of f with respect to m is defined by:

(2.1.1)

∫ f dm =

Σi=1

α_im(A_i)

Remark 2.1.

(Convention)In definition (1.1), the product α_im(A_i) is zero when- ever α_i = 0, even if m(A_i) = +∞.
Theclass of real-valued elementary functions is denoted by E(Ω, A, R) and E+(Ω, A, R) stands for the subclass of non-negative functions of E(Ω, A, R).
Asan elementary function, f has various expressions, however, Definition (1.1) is coherent (i.e, f dm does not depend on one partic- ular expression of f).

In Definition (1.1) we are using an expression of f in which the coefficientα_i are disjoint, called canonical representation of f.

f is well defined,

In fact, for ω ∈ Ω, ∃ !i₀ : ω ∈ A_i0 . So f (ω) = α_i0 . Moreover, since the expression of f is the canonical one, α_i is unique.

CHAPTER THREE

Integration with respect to a measure on Banach spaces in general

In this part, we are going to construct an integral of functions with values in a Banach space (E, +, ∗, ǁ.ǁ_E) over R.

The construction will consist of repeating Step 1M (from the construction of the integral of Real-valued mappings) and defining a new step to replace both Step 2M and Step 3M.

In fact, we are replacing Step 2M ans Step 3M by one new step because they require an order that E need not to have. Moreover, for an E-valued function f , we are not certain of getting a sequence of elementary func- tions that converge to f .

Remark 3.1. : Here we are considering bounded measure in the con- struction of the integral, unless we have corresponding notions of infinity.

The construction of the integral

We are going to construct the Bochner integral of a measurable function

f : (Ω, A, m) → E in two steps.

CHAPTER FOUR

Integration of mappings with respect to a measure on lattice spaces

In this part we are going to discuss the construction of the Bochner in- tegral in Ordered vector spaces. This part is just an introductory part to Integration in Lattice spaces.

Another view on the construction of the Bochner integral

Ordered Banach spaces are Banach spaces by definition, so in this setion we are just recalling the construction done in in chapter3, we will also recall some of the properties of the Bochner integral.

Let us recall the construction of the Bochner integral of functions with values in Banach spaces.

Consider the measure space(Ω, A, m) with m a bounded measure. Consider the Banach space (E, +, ∗, ǁ.ǁ_E) over R

We constructed the Bochner integral of a measurable function f : (Ω, A, m) →

E in two steps.

CHAPTER FIVE

Conclusion and Perspectives

Using the knowledge of Measure Theory, the integration of real-valued measurable mappings can be extended to an integration of measurable mappings with values in Banach spaces, called Bochner Integration.

On R the Bochner integral and the Modern Integral coincide when using bounded measure.

We have been able to establish limit theorems for Banach-valued mea- surable mappings and we also establish an important result which is the Dominated Convergence theorem on Banach spaces in general.

In fact, the integration in linear spaces goes back toBochner(1933),Dunford(1936a),Dunford(1936b),Birkoff(1935),Birkoff(1937).

But these works are summarized inPettis(1938) whose paper is consid- ered as the seminal introduction to vector valued integration. Currently, the topic of integration in Banach spaces, locally convex spaces and in other abstract spaces is very popular, and this thesis is part of this trend.

CONCLUSION AND PERSPECTIVE

Moreover, it is important to emphasize on the fact that this thesis fully discussed the notion of Bochner integration in general Banach Spaces; and also extended some well-know theorems about real-valued functions to functions with values in Banach spaces. Also, since the last chapter was just an introductory part to integration in Lattice spaces, We discussed the notion of ordered Banach spaces and gave the properties on the Bochner integral on Banach Lattice spaces.

As perspectives, this thesis can be more complete by discussing in depth Bochner integration in Lattice spaces and Locally convex spaces. Random set integration on Banach spaces using Bochner integrals and applications is an interesting research topic after reading this thesis that gives us a good idea of Bochner integration.

Bibliography

Birkoff G. (1937). Moore-Smith convernvence in general topology, Annals of Mathematics, (2), Vol. 38, pp 33-57
Birkoff G.(1935). Integration of functions with values in a Banach spaces. Transactions of the American Mathematical Society. Vol 28, pp. 357-378 Bochner S. (1933). Integration von Funktionen, deren die Elemente einses Vektorra˝ umes sind. Funfamenta Mathematicae, Vol 20 (1993), pp 262-
Dunford N. Integration of vector-valued functions. Bulletin of Mathematical Society, abstract, 43-1-21
Dunford N.(1936) Integration and liear operations. Transactions of the American Mathematical Society, Vol 40, pp. 474-494
Eduard Yu. Emelyanov (2013). Note on Archimedean property in ordered vector spaces. arXiv: 1309.2903v1
Lo, G.S.(2018). Measure Theory and Integration by and for the learner. SPAS Editions. Saint-Louis, Calgary, Abuja. Doi : http://dx.doi.org/10.16929/sbs/2016.0005, ISBN : 978-2-9559183-5- 7.

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