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LEBESGUE Integral
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Title page
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Measurable functions
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany
Measurable sets
[chap:MeasurableSets]
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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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Assumptions
In the following S is a given set
and B a given ring of subsets of S .
Moreover, μ : B → [0, ∞[ is an additive measure
(see sect:1-1),
and μ* is the corresponding outer measure
(see sect:1-5).
We assume that μ*=μ on B
(siehe eq:1-essential),
so that the construction of LEBESGUE's integral
in the previous section is applicable.
Let L(μ;Y) be any function space characterized by
axioms axiom:0-axiom:5.
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Note, that in this section we shall only use the axioms
(see axiom:0-axiom:5) for LEBESGUE's integral.
Nothing about the statements in
lemma:1-11-sect:1-15 will be used.
This makes totally clear, that all statements and theorems
in integration theory, without any exception,
can be derived from a few basic properties,
which we have called axioms in sect:1-8.
The important consequences shall be:
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Characterization of convergence in L(μ;Y)
(see theorem satz:2-3).
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The "Theorem of monotone convergence"
(see theorem satz:2-6).
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Extension of the measure μ (using the exterior measure μ* )
on a ϭ -ring (see theorem satz:2-9).
We start with a statement about the connection of
convergence in L(μ;Y) and the "convergence in measure".
Consider a sequence of functions in L(μ;Y) converging in measure
towards a limit function f . Then, in general, it does not follow,
that this sequence converges μ -almost everywhere (pointwise) towards f .
For an example we refer to exercise aufgabe:11.
However, there is always a subsequence converging
μ -almost everywhere towards f .
This is the content of the following proposition.
As next step we carry over properties about step functions from proposition
sect:1-4 to integrable functions in L(μ;Y) :
For the LEBESGUE measure on Rn one can show,
that continuous functions are integrable, provided
they vanish outside a bounded set.
Now comes the central theorem of this section.
Theorem of monotone convergence [satz:2-6]
Let fk ∈L(μ;R) , f : S → R , so that for
μ -almost all x∈S :
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fk(x) ≦fk+1(x)
and
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fk(x) = f(x) .
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Then either
as k → ∞ , or
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f ∈L(μ;R), fk → f in L(μ;R)
and
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fkdμ →
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fdμ |
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as k → ∞ .
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Remark:
The proof shows that the same holds for monotone decreasing sequences.
That is, if fk+1≦fk almost everywhere, then either
or f∈L(μ;R) , fk → f in L(μ;Y) and
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This theorem shall be used during this lecture at several places.
Here it will be essential to construct
a unique extension of the measure μ .
For this we consider instead of sets E⊂S
the corresponding characteristic function ΧE ∈L(μ;R) .
We shall convert set properties into function properties and vice versa.
As a result it turns out, that
{E⊂S ; ΧE ∈L(μ;R)}
essentially is a ϭ -ring, on which
μ can be extended to a ϭ -additive measure.
In partucular we are interested into an application of this
general result to LEBESGUE's measure on Rn .
The following class of sets will play an important role:
Definition (BOREL sets)
Let X be a topological space and O the system of open
subsets of X . The smallest ϭ -algebra A containing
O as subsystem is called the
ϭ -algebra of BOREL sets of X .
The elements of A are called
BOREL sets.
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Erläuterung:
In this general definition the notion of a topologal space
is used, which in Analysis III is not yet available.
(A Topology on a set X is nothing else, than the description
of a system of subsets, called system of "open sets",
with the property, that finite intersections and arbitrary unions
of open sets again are open sets.)
In this lecture it suffices to consider the case X=Rn .
As known from Analysis II, a subset U⊂Rn is open,
if and only if for every point x∈U there is a ball around x
contained in U .
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Using this we obtain the following result.
LEBESGUE measure on Rn [sect:2-10]
By the previous theorem satz:2-9 we are able to extend
the elementary LEBESGUE measure μ in sect:1-2-(iii)
to a complete measure ext(μ)
and call it the
the LEBESGUE measure
, denoted by
Sets in ext(B) are called
LEBESGUE measurable
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It follows that ext(B) is the smallest
ϭ -ring containig all box sets and all
LEBESGUE null sets.
For this LEBESGUE measure
the following holds:
- [sect:2-10-(i)]
All closed sets and all open sets are
LEBESGUE-measurable. Consequently, ext(B)
contains all BOREL sets of Rn .
- [sect:2-10-(ii)]
Regularity of LEBESGUE's measure.
If E is LEBESGUE measurable with Ln(E)<∞ ,
then for every ε>0 there exists
a compact set K and an open set U so, that
K⊂E⊂U and Ln(U∖K)≦ε .
- [sect:2-10-(iii)]
If E⊂Rn is LEBESGUE measurable, then there
exists a BOREL set B⊂Rn with ΧE = ΧB
Ln -almost everywhere. In other words: There is a
Ln -null set N with E∖N = B∖N .
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Remark:
By sect:2-10-(iii), the system ext(B) is
the smallest ϭ -ring containing all BOREL sets and all
LEBESGUE null sets.
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Résumé
At the beginning of this lecture, in chapter chap:LebesgueIntegral,
we started with a ring B
of subsets of S and a
ϭ -additive measure μ : B → [0,∞[
(or equivalently (see eq:1-essential):
an additive measure on B with μ* = μ on B ).
It has been shown, that there exists a corresponding space
L(μ;Y) of Y -integrable functions and an integral
As a consequence, in this chapter chap:MeasurableSets,
we constructed an extension of (S,B,μ) to a
measure space (S,ext(B),ext(μ)) .
Both, L(μ;Y) und
(S,ext(B),ext(μ)) , are uniquely determined.
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Version 1.7
H.W. Alt - 02.01.2007
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LEBESGUE Integral
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Title page
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Measurable functions
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Index |
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany