# Jordan decomposition

Let $(\mathrm{\Omega},\mathcal{S},\mu )$ be a signed measure space, and let $(A,B)$ be a Hahn decomposition for $\mu $. We define ${\mu}^{+}$ and ${\mu}^{-}$ by

$${\mu}^{+}(E)=\mu (A\cap E)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mu}^{-}(E)=-\mu (B\cap E).$$ |

This definition is easily shown to be independent of the chosen Hahn decomposition.

It is clear that ${\mu}^{+}$ is a positive measure^{}, and it is called the *positive variation* of $\mu $. On the other hand, ${\mu}^{-}$ is a positive finite measure, called the *negative variation* of $\mu $.
The measure $|\mu |={\mu}^{+}+{\mu}^{-}$ is called the *total variation ^{}* of $\mu $.

Notice that $\mu ={\mu}^{+}-{\mu}^{-}$. This decomposition of $\mu $ into its positive and negative parts is called the *Jordan decomposition* of $\mu $.

Title | Jordan decomposition |
---|---|

Canonical name | JordanDecomposition |

Date of creation | 2013-03-22 13:27:02 |

Last modified on | 2013-03-22 13:27:02 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 9 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 28A12 |

Defines | positive variation |

Defines | negative variation |

Defines | total variation |