實線上에서 定義된 測度와 函數에 관한 硏究 = A STUDY ON MEASURES AND FUCTIONS ON THE REAL LINE

다국어 초록 (Multilingual Abstract)

The measure is to be understood as the notation which generalizes the length of a line segment, the area of a plane surface or the content of a volume in space, as well as the total amount of mass contained within a certain volume.
It is one of the essential features of measure that it is additve: the measure of the union of a finite number of disjoint sets is the sum of the measures of the separate sets.
We restrict ourselves here to measures for which this is still true for a countable number of disjoint sets.
I shall prove in this paper that exists a one-one correspondence between the collection of all measures ν, 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 will be shown that ν is Lebesgue absolutely continuous if and only if the corresponding g(x) is the integral (between 0 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.