集合函數的全變差

本論文旨在推廣複測度的全變差以及推廣符號測度的約當分解。在第一節中,我們將一個可測集合的複測度,用包含於該可測集合中的鏈來表示,以代替用該可測集的分割來表示。這樣的表示法可以用來推廣全變差的概念到某些對應域為賦範空間的集合函數。在第二節中,我們建立全變差範數的概念,同時介紹賦範空間BV(X,β,Y),這是一個由那些定義域為β (β是一個含空集合以及X的任何一個X的子集合族) 對應域為賦範空間Y的所有函數所構成的賦範空間。我們也討論了該空間的完備性。在第三節中,我們將空間Y限制為實數系,並且將一個有界變差函數表示成為兩個單調函數的差。在本論文最後,我們證明了一個實有界變差函數及其全變差範數的特性化結果。

關鍵字

全變差

並列摘要

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.