View source for Luzin N property

In [[mathematics]], the '''Lebesgue differentiation theorem''' is a theorem of [[real analysis]], which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for [[Henri Lebesgue]].

==Statement==
For a [[Lebesgue integrable]] real or complex-valued function ''f'' on '''R'''<sup>''n''</sup>, the indefinite integral is a [[set function]] which maps a measurable set ''A''&thinsp; to the Lebesgue integral of <math>f \cdot \mathbf{1}_A</math>, where <math>\mathbf{1}_{A}</math> denotes the [[Indicator function|characteristic function]] of the set ''A''. It is usually written 

::<math>\int_{A}f\ \mathrm{d}\lambda,</math> 
with ''λ'' the ''n''&ndash;dimensional [[Lebesgue measure]].

The ''derivative'' of this integral at ''x'' is defined to be 

::<math>\lim_{B \rightarrow x} \frac{1}{|B|} \int_{B}f \, \mathrm{d}\lambda,</math> 

where |''B''| denotes the volume (''i.e.'', the Lebesgue measure) of a [[ball (mathematics)|ball]] ''B''&thinsp; centered at ''x'', and ''B''&nbsp;→ ''x'' means that the diameter of ''B''&thinsp; tends to zero.<br /> 
The ''Lebesgue differentiation theorem'' {{harv|Lebesgue|1910}} states that this derivative exists and is equal to ''f''(''x'') at [[almost everywhere|almost every]] point ''x''&nbsp;∈ '''R'''<sup>''n''</sup>. In fact a slightly stronger statement is true. Note that: 

::<math>\left|\frac{1}{|B|} \int_{B}f(y) \, \mathrm{d}\lambda(y) - f(x)\right| = \left|\frac{1}{|B|} \int_{B}f(y) -f(x)\, \mathrm{d}\lambda(y)\right| \le \frac{1}{|B|} \int_{B}|f(y) -f(x)|\, \mathrm{d}\lambda(y).</math>
 
The stronger assertion is that the right hand side tends to zero for almost every point ''x''. The points ''x'' for which this is true are called the '''[[Lebesgue point]]s''' of ''f''.

A more general version also holds. One may replace the balls ''B''&thinsp; by a family <math>\mathcal{V}</math> of sets ''U''&thinsp; of ''bounded eccentricity''. This means that there exists some fixed ''c''&nbsp;> 0 such that each set ''U''&thinsp; from the family is contained in a ball ''B''&thinsp;  with <math>|U| \ge c \, |B|</math>.  It is also assumed that every point ''x''&nbsp;∈ '''R'''<sup>''n''</sup> is contained in arbitrarily small sets from <math>\mathcal{V}</math>. When these sets shrink to ''x'', the same result holds: for almost every point ''x'',
::<math> f(x) = \lim_{U \rightarrow x, \, U \in \mathcal{V}} \frac{1}{|U|} \int_U f \, \mathrm{d}\lambda.</math>

The family of cubes is an example of such a family <math>\mathcal{V}</math>, as is the family <math>\mathcal{V}</math>(''m'') of rectangles in '''R'''<sup>2</sup> such that the ratio of sides stays between ''m''<sup>&minus;1</sup> and ''m'', for some fixed ''m''&nbsp;≥&nbsp;1. If an arbitrary norm is given on '''R'''<sup>''n''</sup>, the family of balls for the metric associated to the norm is another example.

The one-dimensional case was proved earlier by {{harvtxt|Lebesgue|1904}}. If ''f'' is integrable on the real line, the function
:<math>F(x) = \int_{-\infty}^x f(t) \, \mathrm{d} t</math>
is almost everywhere differentiable, with <math>F'(x) = f(x).</math>

==Proof==
The theorem in its stronger form—that almost every point is a Lebesgue point of a [[locally integrable function]] ''f''—can be proved as a consequence of the [[Lp space#Weak Lp|weak&ndash;''L''<sup>1</sup>]] estimates for the [[Hardy–Littlewood maximal function]].  The proof below follows the standard treatment that can be found in {{harvtxt|Benedetto|Czaja|2009}}, {{harvtxt |Stein|Shakarchi|2005}},  {{harvtxt |Wheeden|Zygmund|1977}} and {{harvtxt|Rudin|1987}}.

Since the statement is local in character, ''f'' can be assumed to be  zero outside some ball of finite radius and hence integrable. It is then sufficient to prove that the set 
:<math>E_\alpha = \Bigl\{ x \in \mathbf{R}^n :\limsup_{|B|\rightarrow 0, \, x \in B}  \frac{1}{|B|} \int_B |f(y)-f(x)| \, \mathrm{d}y  > 2\alpha \Bigr\}</math>
has measure 0 for all ''α''&nbsp;>&nbsp;0. 

Let ''ε''&nbsp;>&nbsp;0 be given. Using the [[Dense set|density]] of [[continuous functions]] of [[Compact set|compact]] [[Support of a function|support]] in [[Lp space#Dense subspaces|''L''<sup>1</sup>('''R'''<sup>''n''</sup>)]], one can find such a function ''g'' satisfying
:<math>\|f - g\|_{L^1} = \int_{\mathbf{R}^n} |f(x) - g(x)| \, \mathrm{d}x < \varepsilon.</math> 
It is then helpful to rewrite the main difference as 
:<math> \frac{1}{|B|} \int_B f(y) \, \mathrm{d}y - f(x) = \Bigl(\frac{1}{|B|} \int_B \bigl(f(y) - g(y)\bigr) \, \mathrm{d}y \Bigr) + \Bigl(\frac{1}{|B|}\int_B g(y) \, \mathrm{d}y - g(x) \Bigr)+ \bigl(g(x) - f(x)\bigr).</math>
The first term can be bounded by the value at ''x'' of the maximal function for ''f''&nbsp;&minus;&nbsp;''g'', denoted here by <math>(f-g)^*(x)</math>:
:<math> \frac{1}{|B|} \int_B |f(y) - g(y)| \, \mathrm{d}y  \leq \sup_{r>0} \frac{1}{|B_r(x)|}\int_{B_r(x)} |f(y)-g(y)| \, \mathrm{d}y = (f-g)^*(x).</math>
The second term disappears in the limit since ''g'' is a continuous function, and the third term is bounded by |''f''(''x'')&nbsp;&minus; ''g''(''x'')|. For the absolute value of the original difference to be greater than 2''α'' in the limit, at least one of the first or third terms must be greater than ''α'' in absolute value. However, the estimate on the Hardy–Littlewood function says that
:<math> \Bigl| \left \{ x : (f-g)^*(x) > \alpha \right \} \Bigr| \leq \frac{A_n}{\alpha} \, \|f - g\|_{L^1} < \frac{A_n}{\alpha} \, \varepsilon,</math>
for some constant ''A<sub>n</sub>'' depending only upon the dimension ''n''.  The [[Markov inequality]] (also called Tchebyshev's inequality) says that 
:<math> \Bigl|\left\{ x : |f(x) - g(x)| > \alpha \right \}\Bigr| \leq \frac{1}{\alpha} \, \|f - g\|_{L^1} < \frac{1}{\alpha} \, \varepsilon</math>
whence 
:<math> |E_\alpha| \leq \frac{A_n+1}{\alpha} \, \varepsilon.</math>
Since ''ε'' was arbitrary, it can be taken to be arbitrarily small, and the theorem follows.

==Discussion of proof==

The [[Vitali covering lemma]] is vital to the proof of this theorem; its role lies in proving the estimate for the Hardy-Littlewood maximal function. 

The theorem also holds if balls are replaced, in the definition of the derivative, by families of  sets with diameter tending to zero satisfying the ''Lebesgue's regularity condition'', defined above as ''family of sets with bounded eccentricity''. This follows since the same substitution can be made in the statement of the Vitali covering lemma.

==Discussion==

This is an analogue, and a generalization, of the [[fundamental theorem of calculus]], which equates a [[Riemann integrable]] function and the derivative of its (indefinite) integral. It is also possible to show a converse - that every differentiable function is equal to the integral of its derivative, but this requires a [[Henstock-Kurzweil Integral|Henstock-Kurzweil]] integral in order to be able to integrate an arbitrary derivative.

A special case of the Lebesgue differentiation theorem is the [[Lebesgue density theorem]], which is equivalent to the differentiation theorem for characteristic functions of measurable sets. The density theorem is usually proved using a simpler method (e.g. see Measure and Category).

==See also==

* [[Differentiation of integrals]]

==References==
* {{cite book
| last = Lebesgue
| first = Henri
| authorlink=Henri Lebesgue
| title = Leçons sur l'Intégration et la recherche des fonctions primitives
| publisher = Gauthier-Villars
| place=Paris
| year = 1904
}}
* {{cite journal
| last = Lebesgue
| first = Henri
| authorlink=Henri Lebesgue
| title = Sur l'intégration des fonctions discontinues|url=http://www.numdam.org/item?id=ASENS_1910_3_27__361_0
| journal = Annales scientifiques de l'École Normale Supérieure
| volume = 27
| year = 1910
| pages = 361–450
| ref = harv 
}}
* Measure and Integral - An introduction to Real Analysis, Richard L. Wheeden & [[Antoni Zygmund]], Dekker, 1977
* Measure and Category, John C. Oxtoby, Springer-Verlag, 1980
* {{Cite book
| last1 = Stein
| first1 = Elias M.
| authorlink=Elias M. Stein
| last2 = Shakarchi
| first2 = Rami
| title = Real analysis
| series = Princeton Lectures in Analysis, III
| publisher = Princeton University Press
| location = Princeton, NJ
| year = 2005
| pages = xx+402
| isbn = 0-691-11386-6
| ref = harv
| postscript = <!--None-->
}} {{MathSciNet|id=2129625}}
*{{cite book |title=Integration And Modern Analysis |series=Birkhäuser Advanced Texts |last=Benedetto |first=John J. |last2=Czaja |first2=Wojciech |publisher=Springer |year=2009 |isbn=0817643060 |pages=361–364}}
*{{cite book |last=Rudin |first=Walter |title=Real and complex analysis |edition=3rd |year=1987  |series=International Series in Pure and Applied Mathematics |publisher=McGraw-Hill |isbn=0070542341}}

{{DEFAULTSORT:Lebesgue Differentiation Theorem}}
[[Category:Theorems in real analysis]]
[[Category:Theorems in measure theory]]