| Summary: | 碩士 === 中原大學 === 應用數學研究所 === 93 === The purpose of this thesis is to generalize the notion of total variation measures of complex measures and the Jordan decompositions of signed measure. In section 1, we represent the complex measure of a measurable set by considering its measurable chains other then its partitions and such a representation can be used to generalize the notion of total variation measures to some class if set functions whose ranges contained in a normed space. In section 2, we define the total variation norm and introduce the normed space BV(X, β, Y ) consisting of those bounded variational functions of β into a normed space
Y where β is a class of subsets of X such that
Ø in β and X in β; the completeness of
BV(X, β, Y ) is also discussed. In section 3, we restrict Y to the real number system and
decompose a bounded variational function as a difference of two monotone functions. We
conclude this thesis by a characterization of real bounded variational functions and their
total variation norms.
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