# Lebesgue decomposition

The Lebesgue decomposition of a function of bounded variation is a canonical representation of this function as a sum of at most three terms. If is a function of bounded variation on the interval , then it can be represented in the form

where is an absolutely-continuous function (see Absolute continuity), is a singular function and is a jump function. In certain cases, for example, if , this representation is unique. The Lebesgue decomposition was established by H. Lebesgue (1904) (see [1]).

#### References

[1] | H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928) |

[2] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) |

[3] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |

*B.I. Golubov*

The Lebesgue decomposition of a -finite signed measure defined on a measurable space ( is a -algebra), with respect to a -finite signed measure defined on this space, is a representation of in the form , where and are -finite signed measures, is absolutely continuous (cf. Absolute continuity) with respect to , and is singular with respect to (cf. Mutually-singular measures). Such a representation is always possible and unique.

#### References

[1] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) |

[2] | N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) |

**How to Cite This Entry:**

Lebesgue decomposition.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_decomposition&oldid=13177