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

Multiple integrals
[chap:MultipleIntegrals]


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In this chapter we prove essentially to central theorems, which are very helpful for the computation of integrals in EUKLIDean spaces. On one side this is the theorem of FUBINI (theorem satz:fubini), on the other side the transformation theorem (theorem satz:4-12). The first one expresses the integral of a product measure in terms of the measure of its products. The second describes, how an integral behaves under a ransformation of the independent variables. Both theorems concern relatively obvious questions about integrals, and also the answers are not surprising. However, the proofs are nontrivial. In particular, it follows from the transformation theorem, that the volume of a set in  Rn  is invariant under rotation, a fact, which intuitively one would exspect from a notion for a volume. As we shall see, the proof of this fact mathematically involves a nontrivial argumentation.

Before we formulate the theorem of FUBINI we consider, as motivation, two examples for its application:

Assumptions for the theorem of FUBINI    [sect:4-1]
Let  (S1, B1, μ1)  and  (S2, B2, μ2)  be measure spaces, which are generated by rings  B10⊂B1  and  B20⊂B2 . Definie a measure space  (S,B,μ)  with  S := S1×S2 , where  B  and  μ  are generated by the elementary product measure  μ1×μ2  (see sect:1-2-(iv) and satz:2-9). We then also write  μ=μ1×μ2 .
Special case: Let  μ1=Ln1  and  μ2=Ln2 . Then we identify, for  n:=n1+n2 , the spaces  Rn  and  Rn1×Rn2  by
(x1,...,xn1,y1,...,yn2) |→ ((x1,...,xn1),(y1,...,yn2)) .
In principle, also othe identifications are possible using other orderings of the coordinate. For example, for  i∈{1,...,n}  the space  Rn  canbe identified with  R1×Rn-1  by
(x1,...,xn) |→ (xi,(x1,...,xi-1,xi+1,...,xn)) .
We mention, that such different oorderings do not change the produkt formela, since for parallelepipedida  E1⊂Rn1  and  E2⊂Rn2  always
Ln(E1×E2) = Ln1(E1)·Ln2(E2) .

Theorem of FUBINI    [satz:fubini]
Let  μ=μ1×μ2  with the assumptions in sect:4-1 and  f∈L(μ;Y) . Then
für μ1-fast alle x1∈S1 :   f(x1,·) ∈L(μ2;Y) ,
für μ2-fast alle x2∈S2 :   f(·,x2) ∈L(μ1;Y) ,
and
( x1  |→ 
 
 
∫
S1
 f(x1,x2)dμ2(x2) )
∈L(μ1;Y) ,
( x2  |→ 
 
 
∫
S1
 f(x1,x2)dμ1(x1) )
∈L(μ2;Y)
with the identity
 
∫
S
 f(x1,x2)dμ(x1,x2)
=
 
 
∫
S1
 (
 
 
∫
S2
 f(x1,x2)dμ1(x1) ) dμ2(x2)
=
 
 
∫
S2
 (
 
 
∫
S1
 f(x1,x3)dμ2(x2) ) dμ1(x1) .
Conclusion 1: Application to the function  |f| ∈L(μ;R)  gives
|| f || L(μ)
=
 
 
∫
S1
 || f(x1,·) || L(μ2)dμ1(x1)
=
 
 
∫
S2
 || f(·,x2) || L(μ1)dμ2(x2) .
Conclusion 2: Application to the characteristic function  f = ΧN   gives: If  N  is a  μ -null set, then we have for  μ1 -almost all  x1 ∈S1 
μ2({x2 ∈S2;  (x1,x2)∈N})=0,
and for  μ2 -almost all  x2 ∈S2 
μ1({x1 ∈S1;  (x1,x2)∈N})=0.

Now we present sone examples, which show how th theorem of FUBINI can be used to compute special integrals.

One of the most important constructions using the LEBESGUE integral in  Rn  is the convolution of two integrable functions. For the proof, that this convolution is well defined, the theorem of FUBINI is used. The convolution is often used, to approximate functions by "smooth functions" (usually infinite differentiable functions) with respect to the integral norm.

The second part of this chapter deals with the transformation formula satz:4-12. For the proof we need two preparating lemmata. In lemma:4-10 we see, that images of LEBESGUE-null sets under local LIPSCHITZ continuous transformations are again LEBESGUE null sets. The second lemma lemma:4-11 confirms the intuitive idea of the influence affine linear mappings to the LEBESGUE measure of measurable sets.

The second lemma is the transformation rule for linear transformations, where we restrict ourselves as a preliminary step to the transformation of a special class of sets.

Lemma    [lemma:4-11]
Let  τ:Rn → Rn  be an affine mapping with
τ(y) = Ay+b ,     A∈Rn×n invertible and b∈Rn.
Then for all compakt convex subset  K∈Rn 
Ln(τ(K)) = | detA|·Ln(K).
In particular, this holds for all closed prallelepipeds.
Remark: By assumption   detA≠0 . Otherwise the image of  τ  is a lower dimensional set, hence a  Ln -null set (see exercise aufgabe:5). Also in this case statement of the lemma is true.

After these preparations we are ready to prove the following transformation theorem for the LEBESGUE integral on  Rn .

Transformation theorem    [satz:4-12]
Let  U, V⊂Rn  be open sets and  τ:U → V  a  C1 -diffeomorphism. Then
f∈L1(V;Y)      <=>     f∘τ| det(Dτ)|∈L1(U;Y) ,
and then the identity (called transformation rule or transformation formula)
[eq:4-12]
 
∫
V
 f(x) dx =
 
∫
U
 (f∘τ)(y)| detDτ(y)| dy
holds, where  Dτ(y)  is the derivative of  τ  at  y .
Remark: By   detDτ(y)  we mean the determinant of the matrix
(ei•Dτ(y)(ej))i,j=1,...,n ,
where  ei ,  i=1,...,n  are the canonical basis vectors of  Rn .
Now we apply the transformation rule to concrete transformations. This allows us to compute some special integrals in  Rn .

Version 1.7
H.W. Alt - 02.01.2007

Measurable functions Measurable functions
Title page Title page
Surfaces in  Rn  Surfaces in  Rn 
Surfaces in  Rn  Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany