Hahn-Jordan decomposition
In measure theory , a branch of mathematics that deals with the generalization of volume concepts, the Hahn-Jordan decomposition describes how a signed measure can be broken down into a negative and a positive part. Sometimes the decomposition is also given as two separate statements; they are then called Hahn's decomposition theorem and Jordan’s decomposition theorem . The two sentences are closely related. Hahn's decomposition theorem was proved by Hans Hahn in 1921; the name of the Jordan decomposition theorem refers to Marie Ennemond Camille Jordan , who showed in 1881 that a function of limited variation can be represented as the difference between two monotonically increasing functions.
Hahnscher decomposition theorem
statement
Be a measuring room and a signed dimension on this measuring room.
Then there is a partition of the basic set into a positive set and a negative set , i.e. and .
comment
The decomposition of the basic space is unambiguous except for a zero set. So if there is another Hahn decomposition, then is and . The symmetrical difference denotes .
variation
Using Hahn's decomposition theorem, the variation , the positive variation and the negative variation can be defined. The variation is sometimes also called total variation or total variation . This designation is ambiguous, however, as it is also used in part for the norm constructed from the variation, the total variation norm .
definition
Is a signed measure with Hahn decomposition , it is called
the positive variation of ,
the negative variation of and
the variation of .
Remarks
- Since the Hahn decomposition is unique up to zeros, the above definitions do not depend on the choice of decomposition.
- The index is also called the total variation norm of a signed measure.
- The positive variation and the negative variation are singular to each other.
Jordan decomposition kit
The Jordanian Decomposition Theorem summarizes once again the decomposition of the signed measure. It reads: is a signed measure, so is
and and are singular to each other, so .
literature
- Camille Jordan : Sur la Série de Fourier. In: Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Vol. 92, No. 5, 1881, ISSN 0001-4036 , pp. 228-230, digitized .
- Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .