测度(Measure)

OLS

数学上，测度(Measure)是一个函数，它对一个给定集合的某些子集指定一个数，这个数可以比作大小、体积、概率等等。传统的积分是在区间上进行的，后来人们希望把积分推广到任意的集合上，就发展出测度的概念，它在数学分析和概率论有重要的地位。

测度论是实分析的一个分支，研究对象有σ代数、测度、可测函数和积分，其重要性在概率论和统计学中有所体现。

In mathematics, more specifically measure theory, a measure is intuitively a certain association between subsets of a given set X and the (extended set) of non-negative real numbers. Often, some subsets of a given set X are not required to be associated to a non-negative real number; the subsets which are required to be associated to a non-negative real number are known as the measurable subsets of X. The collection of all measurable subsets of X is required to form what is known as a sigma algebra; namely, a sigma algebra is a subcollection of the collection of all subsets of X that in addition, satisfies certain axioms.

Measures can be thought of as a generalization of the notions: 'length,' 'area' and 'volume.' The Lebesgue measure defines this for subsets of a Euclidean space, and an arbitrary measure generalizes this notion to subsets of any set. The original intent for measure was to define the Lebesgue integral, which increases the set of integrable functions considerably. It has since found numerous applications in probability theory, in addition to several other areas of academia, particularly in mathematical analysis. There is a related notion of volume form used in differential topology.

杜增

作为一个学物理的学生，我想数学上的“测度”是无法不应该知道的概念，他什么意义，相应的物理含义是什么，我想把它整明白！