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In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.

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  • Mengenfunktion (de)
  • 集合函数 (ja)
  • Set function (en)
  • Функция множеств (ru)
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  • In der Mathematik sind Mengenfunktionen Funktionen, die bestimmten Mengen (den Mengen eines Mengensystems) Werte zuordnen, in der Regel nicht-negative reelle Zahlen oder den Wert . Funktionen, die als Werte Mengen annehmen werden hingegen mengenwertige Funktionen genannt. Mengenfunktionen bilden die Basis für die Maßtheorie, wo unter anderem Mengenfunktionen, wie Maße oder Inhalte, auf genauere Eigenschaften untersucht werden. (de)
  • In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. (en)
  • 数学における集合函数(しゅうごうかんすう、英: set-function)は集合を変数(入力、引数)とする函数である。集合函数は出力としてふつうは数を返すが、しばしば出力として無限大を許す(すなわち補完数直線に値をとる函数も考える)。入力は、普通は適当な集合の部分集合族の元となっているような集合であり、しばしば実数からなる集合、ユークリッド空間内の点集合、適当な測度空間内の点集合などから取られる。 これと対照的に、入力が点である(通常の意味の)函数を点函数とよぶ。また、集合を値として出力する写像はしばしば集合値函数と呼ばれる(集合値函数と多価函数は同じような意味で用いられることがあるが、必ずしも同義語でない)。 集合函数は測度論の基礎を成すもので、測度および有限加法的測度は特定の性質を満足する集合函数として定められる。 (ja)
  • Функция множеств — функция , определённая на некоторой системе множеств с областью значений в некотором множестве . Например, функция , сопоставляющая каждому подмножеству множества его кардинальное число — функция множеств. (ru)
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