absolutely continuous
Let and be signed measures or complex measures on the same measurable space . We say that is absolutely continuous with respect to if, for each such that , it holds that . This is usually denoted by .
Remarks.
If and are signed measures and is the Jordan decomposition of , the following are equivalent:
-
1.
;
-
2.
and ;
-
3.
.
If is a finite signed or complex measure and , the following useful property holds: for each , there is a such that whenever .
Title | absolutely continuous |
---|---|
Canonical name | AbsolutelyContinuous |
Date of creation | 2013-03-22 13:26:12 |
Last modified on | 2013-03-22 13:26:12 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 10 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 28A12 |
Related topic | RadonNikodymTheorem |
Related topic | AbsolutelyContinuousFunction2 |
Defines | absolute continuity |