View source for Lebesgue decomposition

{{MSC|26A45|28A15}}

[[Category:Analysis]]

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===Lebesgue decomposition of a measure===
Called by some authors [[Radon-Nikodym decomposition]]. Consider two $\sigma$-finite nonnegative measures (or [[Signed measure|signed measures]]) $\mu$ and $\nu$ defined on the same [[Algebra of sets|$\sigma$-alegbra]] $\mathcal{B}$ of subsets of $X$. The Lebesgue (or Radon-Nikodym) decomposition is a representation of $\mu$ in the form $\alpha +\beta$ where $\alpha$ is [[Absolute continuity|absolutely continuous]] with respect to $\nu$ and $\beta$ is [[Singular measures|singular]] with respect to $\nu$. Such a  representation is always possible and unique. 
See also [[Differentiation of measures]] and [[Absolute continuity]].

===Lebesgue decomposition of a function of bounded variation===
The Lebesgue decomposition of a function of bounded variation is a canonical representation of a right-continuous function $f:I\to\mathbb R$ of [[Function of bounded variation|bounded variation]] (where $I$ is an interval) as a sum of three functions $f_a+f_c+f_j$, which is unique up to constants.

Using the terminology of Lebesgue:
*$f_j$ is a [[Jump function]] and is given by
\begin{equation}\label{e:jump}
f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, ,
\end{equation}
cp. with page 61 of {{Cite|Le}}. Recall that right and left limit always exist for functions of bounded variation and differ in at most countably many points: the sum in \eqref{e:jump} is then either finite or it is a series, and in the latter case it is absolutely convergent (see [[Function of bounded variation]]). The difference $f-f_j$ is a continuous function of bounded variation and is also called ''continuous part'' of $f$ by Lebesgue. $f_j$ can be characterized as the function $g$ with smallest variation such that $f-g$ is continuous (cp. with Remark 1 at page 163 of {{Cite|Le}}).
*$f_c$ is the ''fonction des singularités'' of the continuous part of $f$ and it is defined as the function $h$ with smallest variation such that $f-f^j -h$ is [[Absolute continuity|absolutely continuous]] (cp. with Section III of Chapter VIII in {{Cite|Le}}; this definition ensures the uniqueness of $f_c$ up to constants: to mod out the constant one can impose, for instance, that $f$ vanishes at the left endpoint of $I$). These functions $f_c$ are also called [[Singular function|singular functions]] and can be characterized in a few ways. For instance they are those continuous functions of bounded variation whose classical derivative vanishes almost everywhere, or those continuous functions of bounded varation whose [[Generalized derivative|distributional derivative]] is singular (with respect to the Lebesgue measure); see [[Function of bounded variation]].
*$f_a$ is given by $f-f_j-f_c$ and it is [[Absolute continuity|absolutely continuous]].

====Cantor parts====
De Giorgi and his school use the name ''Cantor part'' for $f_c$, since a typical example is the [[Cantor ternary function]]. For $f_j$ and $f_a$ they use the terminology ''jump part'' and ''absolutely continuous part'', which agrees with the terminology of Lebesgue. See for instance {{Cite|AFP}}.

====Generalization to higher dimension====
Such decomposition has been generalized to functions of bounded variations depending on several variables. However this generalized decomposition is performed at the level of the [[Generalized derivative|distributional derivative]] of the corresponding function. See [[Function of bounded variation]] for more on this topic.

===References===
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|valign="top"|{{Ref|AFP}}||     L. Ambrosio, N.  Fusco, D.   Pallara, "Functions of bounded  variations   and  free  discontinuity   problems". Oxford Mathematical  Monographs. The     Clarendon Press,   Oxford University Press, New York,  2000.       {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|DS}}||     N. Dunford, J.T. Schwartz,  "Linear operators. General theory",    '''1''', Interscience (1958)  {{MR|0117523}} {{ZBL|0635.47001}}
|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|valign="top"|{{Ref|Le}}|| H. Lebesgue,   "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars  (1928).
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|valign="top"|{{Ref|Na}}|| I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)
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