Chapter 36

Integration On ManifoldsYou can do integration on various manifolds by using the Hausdorff measure of an ap-propriate dimension. However, it is possible to discuss this through the use of the Rieszrepresentation theorem and some of the machinery for accomplishing this is interesting forits own sake so I will present this alternate point of view.

36.1 Partitions Of UnityThis material has already been mostly discussed starting on Page 1023. However, that wasa long time ago and it seems like it might be good to go over it again and so, for the sakeof convenience, here it is again.

Definition 36.1.1 Let C be a set whose elements are subsets of Rn.1 Then C is said to belocally finite if for every x ∈ Rn, there exists an open set, Ux containing x such that Ux hasnonempty intersection with only finitely many sets of C.

Lemma 36.1.2 Let C be a set whose elements are open subsets ofRn and suppose ∪C⊇H,a closed set. Then there exists a countable list of open sets, {Ui}∞

i=1 such that each Ui isbounded, each Ui is a subset of some set of C, and ∪∞

i=1Ui ⊇ H.

Proof: Let Wk ≡ B(0,k) ,W0 =W−1 = /0. For each x ∈ H ∩Wk there exists an open set,Ux such that Ux is a subset of some set of C and Ux ⊆Wk+1 \Wk−1. Then since H ∩Wk iscompact, there exist finitely many of these sets,

{Uk

i}m(k)

i=1 whose union contains H ∩Wk. IfH ∩Wk = /0, let m(k) = 0 and there are no such sets obtained.The desired countable list ofopen sets is ∪∞

k=1

{Uk

i}m(k)

i=1 . Each open set in this list is bounded. Furthermore, if x ∈ Rn,

then x ∈Wk where k is the first positive integer with x ∈Wk. Then Wk \Wk−1 is an openset containing x and this open set can have nonempty intersection only with with a set of{

Uki}m(k)

i=1 ∪{

Uk−1i

}m(k−1)i=1 , a finite list of sets. Therefore, ∪∞

k=1

{Uk

i}m(k)

i=1 is locally finite.The set, {Ui}∞

i=1 is said to be a locally finite cover of H. The following lemma givessome important reasons why a locally finite list of sets is so significant. First of all considerthe rational numbers, {ri}∞

i=1 each rational number is a closed set.

Q= {ri}∞

i=1 = ∪∞i=1{ri} ̸= ∪∞

i=1 {ri}= R

The set of rational numbers is definitely not locally finite.

Lemma 36.1.3 Let C be locally finite. Then

∪C= ∪{

H : H ∈ C}.

Next suppose the elements of C are open sets and that for each U ∈ C, there exists a differ-entiable function, ψU having spt(ψU ) ⊆U. Then you can define the following finite sumfor each x ∈ Rn

f (x)≡∑{ψU (x) : x ∈U ∈ C} .1The definition applies with no change to a general topological space in place of Rn.

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