Lebesgue decomposition

2020 Mathematics Subject Classification: Primary: 26A45 Secondary: 28A15 [MSN][ZBL]

Lebesgue decomposition of a measure

Called by some authors Radon-Nikodym decomposition. Consider two $\sigma$-finite nonnegative measures (or signed measures) $\mu$ and $\nu$ defined on the same $\sigma$-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 absolutely continuous with respect to $\nu$ and $\beta$ is 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 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 [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 [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 absolutely continuous (cp. with Section III of Chapter VIII in [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 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 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 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 [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 distributional derivative of the corresponding function. See Function of bounded variation for more on this topic.

References

[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. MR1857292Zbl 0957.49001
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Le]	H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928).
[Na]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)