Integration in Lattice Spaces

 – Integration in Lattice Spaces –

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ABSTRACT  

The goal of this thesis is to extend the notion of integration with respect to a measure to Lattice spaces. To do so the paper is first summarizing the notion of integration with respect to a measure on R.

Then, a construction of an integral on Banach spaces called the Bochner integral is introduced and the main focus which is integration on lattice spaces is lastly addressed. 

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INTRODUCTION  

Integration is a mathematical technique used to find areas, volumes and so many other mathematical measures.

But to make the notion of integration easy to picture, we define it simply as a mean to find area under the curve of a function; and the result of the integration is called the integral of the function.

Therefore, it is not a surprise that, there are different types of integration, since the functions to be integrated have variety of properties.

The first type of integration that comes to our mind, when we talk about area under a curve is the Riemann integration named as Riemann-Stieltjes integration under its general form.

However, we will see that, this integration is applicable to real-valued functions and requires some specific properties that all real-valued functions need not have.

Therefore, we will introduce another type of integration, called the Lebesgue-Stieltjes integration, that will address most of the limits of the Riemann-Stieltjes Integration.

The Lebesgue –Stieltjes integration gives us a mean to compute the integral of a large range of real-valued function with an undeniable property called measurability. So, the Lebesgue-Stieljes Integration is addressed as the integration of real-valued measurable mappings with respect to a ‘measure’.

While the Riemann-Stieljes integration focus on powerful computation tools, the Lebesgue Stieljes theory adresses powerful results of existence, limits theory, integration and differentiation under the integral sign. As such, it is unavoidable in modern analysis.

Now, the goal of this thesis is not only to expand the integration of realvalued measurable mappings to Banach spaces but also to introduce the notion of integration of measurable mappings with values in Lattice spaces. 

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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
Vektorraumes sind. Funfamenta Mathematicae, Vol 20 (1993), pp 262-276

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.

Li S, Ogura Y. and Krenovich V. (2002). Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. SpringerScience+Business Media, B.V. ISBN/ 978-90-481-6139-3 (hard), ISBN
978-94-015-9932-0 (eBook), DOI: 10.1007/978-94-015-9932-0

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