Additive Set Function
Definition
Consider space
and the 
additive class 
.For a finite real-valued function
in set 
,
.
monotonic increasing.
monotonic decreasing.
is monotonic increasing or monotonic decreasing
Upper variation of
:
Lower variation of
:
Total variation of
:
Jordan decomposition theorem
Hahn's decomposition theorem
For
in space 
, there exists 
satisfying
.If
is absolutely continuous,
.If
is singular,
A necessary and sufficient condition for
to be absolutely continuous is
.A necessary and sufficient condition for
to be singular is that
.Radon-Nikodym theorem
- An absolutely continuous set function
and a singular set function 
exist, with
- The above is uniquely defined.
- For
,
where
is an integrable function.
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