Lebesgue differentiation theorem
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lebesgue differentiation theorem is a theorem of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
, 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 Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
.


Statement

For a
Lebesgue integrable In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
real or complex-valued function ''f'' on R''n'', the indefinite integral is a
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ...
which maps a measurable set ''A'' to the Lebesgue integral of f \cdot \mathbf_A, where \mathbf_ denotes the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of the set ''A''. It is usually written A \mapsto \int_ f\ \mathrm\lambda, with ''λ'' the ''n''–dimensional
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. The ''derivative'' of this integral at ''x'' is defined to be \lim_ \frac \int_f \, \mathrm\lambda, where , ''B'', denotes the volume (i.e., the Lebesgue measure) of a
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
''B''  centered at ''x'', and ''B'' → ''x'' means that the diameter of ''B''  tends to 0.
The ''Lebesgue differentiation theorem'' states that this derivative exists and is equal to ''f''(''x'') at
almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
point ''x'' ∈ R''n''. In fact a slightly stronger statement is true. Note that: \left, \frac \int_f(y) \, \mathrm\lambda(y) - f(x)\ = \left, \frac \int_(f(y) - f(x))\, \mathrm\lambda(y)\ \le \frac \int_, f(y) -f(x), \, \mathrm\lambda(y). 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 In mathematics, given a locally Lebesgue integrable function f on \mathbb^k, a point x in the domain of f is a Lebesgue point if :\lim_\frac\int_ \!, f(y)-f(x), \,\mathrmy=0. Here, B(x,r) is a ball centered at x with radius r > 0, and \lambda (B(x ...
s of ''f''. A more general version also holds. One may replace the balls ''B''  by a family \mathcal of sets ''U''  of ''bounded eccentricity''. This means that there exists some fixed ''c'' > 0 such that each set ''U''  from the family is contained in a ball ''B''  with , U, \ge c \, , B, . It is also assumed that every point ''x'' ∈ R''n'' is contained in arbitrarily small sets from \mathcal. When these sets shrink to ''x'', the same result holds: for almost every point ''x'', f(x) = \lim_ \frac \int_U f \, \mathrm\lambda. The family of cubes is an example of such a family \mathcal, as is the family \mathcal(''m'') of rectangles in R2 such that the ratio of sides stays between ''m''−1 and ''m'', for some fixed ''m'' ≥ 1. If an arbitrary norm is given on R''n'', the family of balls for the metric associated to the norm is another example. The one-dimensional case was proved earlier by . If ''f'' is integrable on the real line, the function F(x) = \int_ f(t) \, \mathrm t is almost everywhere differentiable, with F'(x) = f(x). Were F defined by a
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
this would be essentially the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, but Lebesgue proved that it remains true when using the Lebesgue integral.


Proof

The theorem in its stronger form—that almost every point is a Lebesgue point of a
locally integrable function In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies ...
''f''—can be proved as a consequence of the weak–''L''1 estimates for the
Hardy–Littlewood maximal function In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another f ...
. The proof below follows the standard treatment that can be found in , , and . 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 :E_\alpha = \Bigl\ has measure 0 for all ''α'' > 0. Let ''ε'' > 0 be given. Using the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
in ''L''1(R''n''), one can find such a function ''g'' satisfying :\, f - g\, _ = \int_ , f(x) - g(x), \, \mathrmx < \varepsilon. It is then helpful to rewrite the main difference as : \frac \int_B f(y) \, \mathrmy - f(x) = \Bigl(\frac \int_B \bigl(f(y) - g(y)\bigr) \, \mathrmy \Bigr) + \Bigl(\frac\int_B g(y) \, \mathrmy - g(x) \Bigr)+ \bigl(g(x) - f(x)\bigr). The first term can be bounded by the value at ''x'' of the maximal function for ''f'' − ''g'', denoted here by (f-g)^*(x): : \frac \int_B , f(y) - g(y), \, \mathrmy \leq \sup_ \frac\int_ , f(y)-g(y), \, \mathrmy = (f-g)^*(x). The second term disappears in the limit since ''g'' is a continuous function, and the third term is bounded by , ''f''(''x'') − ''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 : \Bigl, \left \ \Bigr, \leq \frac \, \, f - g\, _ < \frac \, \varepsilon, for some constant ''An'' depending only upon the dimension ''n''. The
Markov inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named a ...
(also called Tchebyshev's inequality) says that : \Bigl, \left\\Bigr, \leq \frac \, \, f - g\, _ < \frac \, \varepsilon whence : , E_\alpha, \leq \frac \, \varepsilon. Since ''ε'' was arbitrary, it can be taken to be arbitrarily small, and the theorem follows.


Discussion of proof

The
Vitali covering lemma In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The co ...
is vital to the proof of this theorem; its role lies in proving the estimate for the
Hardy–Littlewood maximal function In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another f ...
. 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 The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, which equates a
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
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 in order to be able to integrate an arbitrary derivative. A special case of the Lebesgue differentiation theorem is the
Lebesgue density theorem In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intuit ...
, 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). This theorem is also true for every finite Borel measure on R''n'' instead of Lebesgue measure (a proof can be found in e.g. ). More generally, it is true of any finite Borel measure on a separable metric space such that at least one of the following holds: * the metric space is a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, * the metric space is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
, * the measure is doubling. A proof of these results can be found in sections 2.8–2.9 of (Federer 1969).


See also

*
Lebesgue's density theorem In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intuit ...


References

* * * * * * * * * {{DEFAULTSORT:Lebesgue Differentiation Theorem Theorems in real analysis Theorems in measure theory