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申鉉千,金容泰 成均館大學校 1962 論文集 Vol.7 No.-
This thesis was devoted to some concepts generalizing the concept of the measure. More specifically, we considered signed measures (difference of two finite measures) and complex masures v'+iv" (where v' and v" are signed measures). Among the most important properties were the facts that the collection of all signed (or complex) measures was a real (or comple) Branch space, and that any (finite) limit of a converging sequence of signed (or complex) measures was a signed (or complex) measure. We proved in this thesis that there existed a one-one correspondence between the collection of all measures v, initially defined (and finite) on the semi-ring of all cells in R_1, and the collection of all functions g(x) on R_1, increasing on R_1 and vanishing at the origin. In addition, it was shown that v was Lebesgue absolutely continuous if and only if the corresponding g(x) was the integral (between o and x) of its derivative g'(x) . In the exercises one may find the Lebesgue decomposition theorem for an increasing function in its original version.