 |
Contents (LaTeX)
|
|
 |
Title page
|
|
 |
Measurable sets
|
|
 |
Index |
|
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
LEBESGUE Integral
[chap:LebesgueIntegral]
|
This is an english version of the script
|
|
|
You may switch to the original german version:
|
|
|
The goal of this first chapter is the development of a general notion
of an integral, called LEBESGUE integral.
This integral is needed in nearly all theories in analysis.
The main subjects of this chapter are:
- Additive measures and integral of primitive functions
- Outer measure and null sets
- Axioms of Lebesgue integral
- Construction of Lebesgue integral
Remark
In this lecture we started, as a motivation,
with some elementary problems,
for which the computation of integrals
is of essential importance:
- Computation of the area of a surface in R2 ,
which is given as the graph of a function.
- Computation of the volume of a body in R3 ,
which is given as a subset.
- Computation of moments of a set of points in Rn .
- Computation of moments of a body in R3 ,
for example its centre.
- Computation of an expectation value.
|
We start with additive measures μ defined on a ring S
and define an elementary integral
on the vector space of step functions T(μ;Y) .
This elementary integral leads, in a natural way, to
a seminorm on T(μ;Y) .
Introducing the notion of μ -null sets and a
corresponding equivalence relation on T(μ;Y) ,
this seminorm becomes a norm.
Then the space L(μ;Y) of
integrable functions ist nothing else than
the completion of the space of step functions
with respect to this norm.
Moreover, it will turn out that the LEBESGUE integral for
integrable functions is the closure of the elementary
integral with respect to the above norm.
As example we introduce some important additive measures.
Some elementary measures [sect:1-2]
|
Based on additive measures we consider so called "step functions"
and introduce an elementary integral for such functions.
Basic properties of this elementary integral
are contained in the following statement.
We have to extend our notions in order to be able to measure
more general subsets of Rn and to integrate more general functions.
We start with a motivation.
This example shows, that it is reasonable to consider coverings
of a given set by countably many sets of the given ring of subsets.
For the development of the theory of integration we
need corresponding definitions.
Moreover, we are lead to the notion of an "outer measure",
which will play an important role for all following considerations.
As consequence of this definition we see, that
we have to deal with classes of functions,
which are invariant under a change of their values on a null set.
This leads to properties which hold "almost everywhere".
Imposing the following condition on the outer measure μ* of μ
we are able to prove a fundamental property for step functions.
This property is essential for the development of
the theory of integration.
As a matter of fact, we shall show that the above property
is satisfied for the LEBESGUE measure on Rn .
Remark
The following statements are equivalent:
- μ*=μ on B .
- μ is ϭ -subadditive on B .
- μ is ϭ -additive on B .
The proof is the content of exercise aufgabe:7.
|
Now we introduce an equivalence relation on the vector space T(μ;Y)
and prove, that with this equivalence relation the space
T(μ;Y) becomes a normed space.
Thus we have seen, that under assumption eq:1-essential
the space T(μ;Y) of step functions with the above equivalence relation
becomes a normed vector space.
However, T(μ;Y) ist not a complete vector space.
The idea of the theory of LEBESGUE integration is,
to consider the completion of the normed space T(μ;Y) .
This completion, which we might denote by L(μ;Y) ,
exists in an abstrakt manner as the set of
CAUCHY sequences in T(μ;Y) .
The goal is to find a characterization of
L(μ;Y) as a function space.
The procedure is as follows:
First we formulate such a characterization in an axiomatic manner.
Then we shall see, that L(μ;Y) is uniquely determined,
moreover, it can be expressed in a natural way in terms of T(μ;Y) .
Axioms of LEBESGUE integral [sect:1-8]
Let S be a set, B a ring of subsets of S ,
μ : B → [0,∞) an additive measure
satifying μ=μ* on the ring B .
Moreover, let Y be a BANACH space
(for the purpose of this lecture one can confine oneself
to the case of an EUKLIDian space Y=Rk , k∈N ).
The LEBESGUE integral consists of:
-
A set L(μ;Y) of
"integrable" functionen mapping S
to Y with the following equivalence relation:
f=g in L(μ;Y) <==> f=g μ -almost everywhere.
-
A mapping f |→ ∫ S f dμ called
"integral". It assigns to each integrable function
f∈L(μ;Y) a value in Y .
- A mapping f |→ || f || L(μ) called
"norm". It assigns to each integrable function
f∈L(μ;Y) a value in R .
We assume that the following properties are satisfieed:
- [axiom:0] Axiom 0.
L(μ;Y) is a vector space
containing T(μ;Y) . Any function,
which conincides with a function in L(μ;Y)
μ -almost everywhere,
belongs to L(μ;Y) .
- [axiom:1] Axiom 1.
f |→ || f || L(μ)
is a norm on L(μ;Y) , and coincides with the
T(μ) -norm on T(μ;Y) .
- [axiom:2] Axiom 2.
f |→ ∫ S f dμ is
linear continuous map with respect to the norm in axiom axiom:1,
and coincides with the elementary integral
on T(μ;Y) .
- [axiom:3] Axiom 3.
For f ∈ L(μ;Y) and ε> 0 we have:
|
|
|| f || L(μ) ≧ε·μ*({x
∈S ; |f(x)| > ε})
|
|
|
|
| Axiom axiom:3: The area
enclosed by the curve is supposed to be greater or equal
the area of the rectangle
{x ; |f(x)| > ε}×[0,ε] [fig:axiom3] |
- [axiom:4] Axiom 4.
L(μ;Y) equipped with the norm
|| · || L(μ) from Axiom axiom:1 is
complete, that is, every CAUCHY sequence in L(μ;Y) has
a limit with respect to this norm.
|
In detail:
For every sequence (fk)k ∈N in L(μ;Y)
satifying || fk- fl || L(μ) → 0 as k,l → ∞
there exists f∈L(μ;Y) with
|| fk - f || L(μ) → 0 as k → ∞ .
|
- [axiom:5] Axiom 5.
T(μ;Y) is dense in
L(μ;Y) , that is, every function in L(μ;Y)
can be approximated by
funtions in T(μ;Y) with respect to the norm in
axiom axiom:1.
|
In detail:
For every f ∈L(μ;Y) there exist
fk ∈T(μ;Y) for k ∈N
satisfying || fk - f || L(μ) → 0 as k → ∞ .
|
|
It turnes out, that the canonical realization of L(μ;Y)
is given by all limits of CAUCHY sequences in
T(μ;Y) , which converge almost everywhere (see satz:1-9 below).
The remainder of this chapter is devoted to the proof,
that this particular space L(μ;Y) satisfies the above axioms.
This proves that LEBESGUE's integral exists.
Existence of LEBESGUE integral [satz:1-9]
Let S , B , μ as in sect:1-8.
Then the axioms in sect:1-8 are satisfied for the space
|
| |
| | |
There exists a CAUCHY sequence (fk)k ∈N in T(μ;Y),
|
|
|
| | |
so that for μ-almost all x ∈S :
|
|
|
| | |
fk(x) → f(x) in Y for k → ∞} |
|
| |
|
|
equipped with the equivalence relation in sect:1-8,
and with the following integral and norm:
For f ∈L(μ;Y) and every sequence (fk)k ∈N
according to f
as in the above definition let
|
| |
| | |
|
| | |
|| f || L(μ)
:=
| |
|| fk || T(μ) .
|
|
| |
|
|
|
Notice:
The existence of these limits as well as the independence
of these limits on the choice of the sequence (fk)k ∈N
is part of the statement.
|
|
First of all let us show,
that our construction is applicable
to the elementary LEBESGUE measure on Rn .
Below we give a detailed proof of theorem satz:1-9.
In the proof we shall often apply the following useful lemma
about the choice of
certain subsequences (this will be needed also in subsequent chapters).
Lemma [lemma:1-10]
Let X be a vector space
with seminorm || · || .
If (xk)k ∈N
is a CAUCHY sequence in X ,
then there exists a subsequence
(xki)i ∈N
with
|| xki+1 - xki || ≦2-i .
|
|
Proof.
Wähle ki mit || xk - xl || ≦2-i für
k,l ≧ki und induktiv in i wähle ki > ki-1 . Damit folgt
die Behauptung.
|
The proof of theorem satz:1-9 is devided into several steps.
Version 1.7
H.W. Alt - 02.01.2007
|
|
Contents (LaTeX)
|
|
|
|
Title page
|
|
|
|
Measurable sets
|
|
|
|
Index |
|
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany