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2015/02/16

measures

A measure m  is a set function which maps a subset in the measure spaces to a real number. 
m: a\in\mathcal{F}\rightarrow m(a)=x\in\mathbb{R}

, where \mathcal{F}  is a family of subsets of the measure space. 

What conditions are needed for a measure m as a set function?

On standard definitions (or finally reached definitions), it is as follows;

(1) for any a\in\mathcal{F} ,  m(a)\geq 0 ,
(2) m(\phi)=0 ,
(3) if a_1,a_2,\cdots\in\mathcal{F}  ,  a_i\cap a_j=\phi (i\ne j) , then m(\cup a_i)=\sum m(a_i)  (i,j=1,2,\cdots

You may think it is very natural. Perhaps it is fairly accurate in terms of  finite operations or various sets having a number of good shapes. 

However, there are very strange sets (or subsets) in a measure space.  

Therefore, these conditions are closely related with the ones which the families \mathcal{F} of subsets satisfy. 






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