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Measurable sets
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Title page
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Multiple integrals
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
Measurable functions
[chap:MeasurableFunctions]
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This is an english version of the script
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You may switch to the original german version:
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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:
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fk continuous,
fk → f locally uniformly
==> f continuous.
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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
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fk satisfies (*),
fk → f μ-almost everywhere
==> f satisfies (*).
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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
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BOREL sets and LEBESGUE null sets,
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Moreover, in the following Y always is an EUKLIDian space, Y=Rl .
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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
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U ⊂Y is an open set
==> f-1 (U) ∈B,
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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
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.
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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.
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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.
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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.
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| 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]
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For the LEBESGUE integral:
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For the space of LEBESGUE integrable functions in Rn :
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For LEBESGUE measurable sets E ⊂Rn :
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L1(E;Y) := {f ∈L(Ln;Y) ;
f(x) = 0 for x ∈Rn ∖E },
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and for f ∈L1(Rn;Y) :
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|| f || L1(E) := || f· ΧE || L(Ln)
=
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| f(x) | dx .
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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
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 :
∫ Rn∖Br(0)ϕk(x) dx → 0 ,
as k → ∞ .
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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
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Measurable sets
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Title page
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Multiple integrals
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|
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany