用户:Råy kuø/沙盒

集合论中,指示函数是定义在某集合X上的函数,表示其中有哪些元素属于某一子集A。指示函数有时候也称为示性函数特征函数

X的子集A的指示函数是函数,定义为

 若
 若

A的指示函数也记作

简单性质

X的子集A对应到它的指示函数的映射是双射,值域是所有函数 的集合。

如果ABX的两个子集,那么

 

以及

 

更一般地,设A1, ..., AnX的子集。对任意 ,可知

 当且仅当x不属于任何Ak

故有

 

展开左式

   
 

其中|F|是F。这是容斥原理的一个形式。

如上一例子所示,指示函数是组合数学一个有用记法。这记法也用在其他地方,例如在概率论:若X概率空间,有概率测度PA可测集,那么1A就是随机变量,其期望值等于A的概率。

 

这等式用于马尔可夫不等式的一个简单证明里。

平均、变异数以及共变异数

给定一个几率空间   ,其中 指示函数可以被定义成以下形式:

   

平均
 
变异数
 
共变异数
 

指示函数的微分

以下我们讨论一个特别的指示函数,黑维塞函数 

The distributional derivative of the Heaviside step function is equal to the Dirac delta function, i.e.   and similarly the distributional derivative of   is  

Thus the derivative of the Heaviside step function can be seen as the inward normal derivative at the boundary of the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain D. The surface of D will be denoted by S. Proceeding, it can be derived that the inward normal derivative of the indicator gives rise to a 'surface delta function', which can be indicated by  :   where n is the outward normal of the surface S. This 'surface delta function' has the following property:[1]  

By setting the function f equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area S.

  1. ^ Lange, Rutger-Jan. Potential theory, path integrals and the Laplacian of the indicator. Journal of High Energy Physics. 2012, 2012 (11): 29–30. Bibcode:2012JHEP...11..032L. S2CID 56188533. arXiv:1302.0864 . doi:10.1007/JHEP11(2012)032.