Hahn decomposition theorem

In measure theory, a branch of mathematics, describes the Hahn- Jordan decomposition, how can a signed measure, ie a measure that can adopt positive or negative values ​​in a negative and a positive part disassemble.

Let be a measurable space and a signed measure it.

A measurable set is called positive with respect to the signed measure, if any measurable subset non-negative measure has, so

Analogy is, a measurable quantity negative with respect to the signed measure, if any measurable subset, not positive measure has, so

The decomposition theorem of Hahn, after the Austrian mathematician Hans Hahn, now states that for every signed measure of a measurable partition of into two disjoint sets, exists, which is with respect to the measure positive with respect to the measure and negative. The partition of the set into a negative and a positive part is called the Hahn decomposition (or Hahnsche decomposition ).

The decomposition is unique modulo null sets, so there is a second decomposition with these properties, the symmetric differences have measure zero:

.

Is thus divided into two dimensions and ordinary ( unsigned ), so

Where and for any measurable amount by

Are defined.

Is called the upper variation, the lower variation. is also a measure, and is referred to as the complete or total variation in the measurement. The representation of the difference between the upper and lower variation called the Jordan decomposition of measurement.

This name refers to Marie Ennemond Camille Jordan, who showed in 1881 that a function of bounded variation can be represented as the difference of two monotone increasing functions.