Inequalities and Set Function Integrals Page: II
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Milligan, Kenneth Wayne, Inequalities and Set Function
Integrals. Master of Science (Mathematics), December, 1971,
This thesis investigates some inequalities and some
relationships between function properties and integral
properties. The material is presented in three chapters:
Introduction and Preliminary Discussion, Field of Sets,
and Integrals. Preliminary definitions and theorems which
are assumed are stated in Chapter I. It is assumed that the
reader is familiar with the elementary properties of the
Riemann-Stieltjes integral and certain basic theorems of
analysis. The logarithm function is defined and some of its
basic properties are demonstrated. Some basic inequalities
are then developed.
The structure of a field of sets and the properties of
functions defined on this field are developed in'chapter II.
The inequalities of the preceding chapter are used to assert
the existence of certain integrals. An important result
presented in this chapter is that if g is a finitely
additive, bounded set function then the integral of the
absolute value of g exists for every set in the field.
The third and final chapter presents the notion of a
continuous function on a field of sets and a norm infimum
function. The relationship between finitely additive,
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Milligan, Kenneth Wayne. Inequalities and Set Function Integrals, thesis, December 1971; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc131479/m1/2/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .