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
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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) |
Lebesgue decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_decomposition&oldid=27820