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?
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.
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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