Hahn-Jordan decomposition

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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

Web links