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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany

Measurable functions
[chap:MeasurableFunctions]


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This chapter deals with the supply of handy criteria for checking the integrability of given functions. For LEBESGUE measurable sets in  Rn  we have proved such a criterium in chapter chap:MeasurableSets: Each such set is the union of a LEBESGUE null set and a BOREL set. So far the only property of an integrable function we know is, that it can be approximated by step functions with respect to the integral norm.

In satz:2-5 we have proved, that all continuous functions in  Rn  with compact support are LEBESGUE integrable, and moreover, in the one-dimensional case  n=1 , also all RIEMANN integrable functions. The question is, whether one can check for a general given function in an convenient way its integrability.

Now, for a continuous functions on  Rn  the following property holds:

fk continuous,  fk → f   locally uniformly      ==>     f continuous.

For the theory of integration, as we have seen during the construction of LEBESGUE's integral in satz:1-9 as well as in the convergence theorem satz:2-3, the notion of almost everywhere convergence is essential. Therefore, in analogy to the notion of continuity, we are searching for a property (*) satisfying

fk satisfies (*),  fk → f  μ-almost everywhere      ==>     f satisfies (*).
As it turns out, this notion is the measurability of a function. The corresponding definition is the content of sect:3-1. We shall show in satz:3-2, that the class of integrable functions is exactly the class of measurable functions  f  for which  |f|  is integrable. This reduces the integrability of general functions to measurability and the integrability of nonnegative functions.

Based on this statement we then develop the theory of integration. Besides the completeness of the space of integrable functions the following theorems belong to the most important statements in the LEBESGUE theory of integration:

As application of the dominated convergence theorem we prove the following frequently used statements about the interchange of limits and integrals:

We finish this section with some applications of these theorems concernning the FOURIER transformation, the convolution of functions and the NEWTON potential.

Assumptions

In the following let  (S,B,μ)  be a measure space satisfying  S ∈B . We assume that  B  is generated by a ring, on which  μ  has finite values (as in satz:2-9).
Standard case: The standard case for which these assumptions are satisfied is
S
=
Rn ,
B
=
the ϭ-ring generated by
BOREL sets and LEBESGUE null sets,
μ
=
Ln on B.
Moreover, in the following  Y  always is an EUKLIDian space,  Y=Rl .

The following definition has an analogy for continuous mappings, which are defined by the property, that the preimage of an open set is an open set.

Measurable functions    [sect:3-1]

(Let  (S,B,μ)  as above and  Y  finite dimensional.) A mapping  f : S → Y  is called  μ -measurable, if
U ⊂Y is an open set ==> f-1 (U) ∈B,
that is, the preimage of open sets is measurable. For this notion of measurability the following holds (all image spaces are assumed to be finite dimensional):
  • [sect:3-1-(i)] Let  fk : S → Y ,  k∈N , be  μ -measurable functions with  fk → f   μ -almost everywhere. Then  f  is  μ -measurable.
  • [sect:3-1-(ii)] Let  f : S → Y  be  μ -measurable and  ϕ : Y → Ỹ  continuous. Then  ϕ∘f  is  μ -measurable.
  • [sect:3-1-(iii)] For mappings  f1 : S → Y1 ,  f2 : S → Y2  the following holds:  f1  and  f2  are  μ -measurable, if and only if  (f1,f2) : S → Y1×Y2  is  μ -measurable.
  • [sect:3-1-(iv)] Let  fk : S → R ,  k∈N , be  μ -measurable. Then also
    g :=
     
    inf
    k∈N
    fk     and     h :=
     
    lim inf
    k → ∞
    fk =
     
    lim
    k → ∞
     
    inf
    j∈N, j≧k
    fk
    are  μ -measurable, provided  g  resp.  h  is finite  μ -almost everywhere. On the null set, where these functions are  -∞ , the values of these functions can be set arbitrarily.

The following theorem provides a characterization of integrable functions through measurability. The result is the majorant criterium for integrability. This is the essential statement of this section. From this all central theorems of the LEBESGUE theory follow. The idea, to reduce the integrability of vector valued functions to the modulus of a function is due to BOCHNER. Therefore, the integral of vector valued functions often is called the BOCHNER-Integral.

Philosophy: In many text books on integration one finds the "traditional" procedure to define the integral of real valued functions by considering lower and upper sums. The integral is the supremum of all lower sums and well defined, if it coincides with the infimum of all upper sums. We plead that this is not the adequate approach.

One reason comes from classical examples like the computation of the centre of gravity of a rigid body. The position vector is a vector valued function, and therefore the integral of the position vector is an integral of a vector valued function. Another example is the computation of the mean value of measurements, if they involve more than one scalar quantity. Advanced examples are the operator theory in Linear Functional Analysis and the treatment of parabolic differential equations as Evolution Equations. Here BANACH space valued integrals are necessary.

Thus, our presentation of LEBESGUE's theory of integration is nothing else than a consequent realization of BOCHNER's idea.

Moreover, we mention that there were always efforts to use a completion principle as basis idea for the theory of integration. The version presented in chap:LebesgueIntegral is based completely on this principle.

Further, in many text books as a preparation step for integration the extension of additive measures is studied. We plead that this is not an efficient approach. Properties of measurable sets are equivalent to statements for the characteristic functions of these sets. Therefore, in our approach in section chap:MeasurableSets the extension of measures is a first consequence of the construction of LEBESGUE's integral.

Theorem (Majorant criterium)    [satz:3-2]

For  f : S → Y  the following is equivalent:
  • [satz:3-2-(i)]  f ∈L(μ;Y) .
  • [satz:3-2-(ii)]  f  is  μ -measurable and  |f| ∈L(μ;R) .
  • [satz:3-2-(iii)] Majorant criterium:   f  is measurable and there exists a function  g ∈L(μ;R)  with  g ≧0 , so that  |f| ≦g   μ -almost everywhere.

-*- FIGURE NOT AVAILABLE -*-
Zum Majorantenkriterium  [fig:majorante]

Let us present some important classes of integrable functions.

As a consequence the central theorems about the LEBESGUE theory drop out.

In the sequel of this lecture statements in particular concern the  n -dimensional LEBESGUE measure  Ln . For this measure special notations are common:

Notations for the LEBESGUE measure    [sect:3-9]

  • For the LEBESGUE integral:
     
     
    Rn
    f dLn
    =
     
     
    Rn
    f(x) dLn(x)
    =
     
     
    Rn
    f(x) dx .
  • For the space of LEBESGUE integrable functions in  Rn :
    L(Ln;Y) = L1(Rn;Y) .
  • For LEBESGUE measurable sets  E ⊂Rn :
    L1(E;Y) := {f ∈L(Ln;Y)  ;  f(x) = 0 for x ∈Rn ∖E },
    and for  f ∈L1(Rn;Y) :
     
    E
    f(x) dx :=
     
     
    Rn
    ΧE ·f(x) dx ,
    || f || L1(E) := ||ΧE || L(Ln) =
     
    E
    | f(x) | dx .

We finish this chapter with some applications of the  n -dimensional LEBESGUE measure.

These applications deal with linear integral operators on  Rn . These are mappings, which assign to a function  f : Rn → R  a function

x  | 
 
 
Rn
K(x,y) f(y) dy .
Here  (x,y) | K(x,y)  is called the kernel of this operator. The following two examples are the most important once.

One of the important applications of convolution in Analysis is the approximation of integrable functions by smooth functions.

For this one considers the convolution of a given function with a sequence of probability distributions  ϕk ≧0 ,  k ∈N , which, for  k → ∞ , more and more are concentrated around the origin. The mathematical formulation of this fact leads to the notion of a DIRAC sequence, which is introduced in the following.

As an application of this procedure see the WEIERSTRASS approximation theorem satz:3-15.

Definition (DIRAC sequence)    [definition:3-12]

A sequence  (ϕk)k ∈N  with  ϕk ∈L1(Rn;R)  is called DIRAC sequence, if
  • [definition:3-12-(i)]  ∀k ∈N :   ϕk ≧0 .
  • [definition:3-12-(ii)]  ∀k ∈N :   ∫ Rnϕk(x) dx = 1 .
  • [definition:3-12-(iii)]  ∀r > 0  :   ∫ RnBr(0)ϕk(x) dx → 0 , as  k → ∞ .

Using the concept of DIRAC sequences we obtain the following approximation lemma.

Another statement, which can be proved using appropriate DIRAC sequences, concerns the approximation of continuous functions by polynomials:

Important applications of the above theorems about the theory of integration concern linear integral operators with kernels having a singularity. The standard example, which plays an inportant role in theoretical physics, ist the


Version 1.7
H.W. Alt - 02.01.2007

Measurable sets Measurable sets
Title page Title page
Multiple integrals Multiple integrals
Multiple integrals Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany