What are Set Functions
What are Set Functions
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DOI：
10.1080/00029890.1948.11991896
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发表时间：
1948
影响因子：
0.5
通讯作者：
A. Rosenthal
中科院分区：
文献类型：
--
作者：
A. Rosenthal
1. Point functions and set functions. In speaking of a function, one usually thinks of a point function f (x): A rule is given by which, in a certain domain, to each number x (or, more generally; to each point x of a certain space) there correspond numbers f (x). If in this statement we replace x by a set X of numbers or points x, we come immediately to the more general notion of a set function f (X); that is: A rule is given by which to each set X, belonging to a certain family of sets of a space, there correspond numbersf (X). In the most important case of a single-valued set function, to each set X only one number f (X) is attachecl. Particular instances of set functions have been well known to mathematicians for a long time, even since antiquity. We have only to remember the length of an interval or of an arc, the area of a plane region or of a surface, the volume of a solid, the content or the measure of a set. Of course, the definite integral is also a very important ex" ample of a set function. In fact, the integral J: f (x) dx is an interval function; to be specific, it is a function of the interval [a, b]. In the case of multiple integrals one integrates over regions. The idea of more general integration over (measurable) sets is due to H. Lebesgue [1]. In fact, it is Lebesgue [2] who may be considered as the founder of the theory of set functions; since his original studies the theory has been further developed by many other mathematicians. We now know that such a general theory is interesting and useful, furnishing a solid, unifying basis for more special theories, and is a valuable source for many applications. A systematic presentation and
