29.7. THE DIVERGENCE THEOREM 1023

29.7 The Divergence TheoremAs an important application of the area formula I will give a general version of the diver-gence theorem for sets in Rp. It will always be assumed p≥ 2. Actually it is not necessaryto make this assumption but what results in the case where p = 1 is nothing more thanthe fundamental theorem of calculus and the considerations necessary to draw this conclu-sion seem unneccessarily tedious. You have to consider H 0, zero dimensional Hausdorffmeasure. It is left as an exercise but I will not present it.

It will be convenient to have some lemmas and theorems in hand before beginningthe proof. First recall the Tietze extension theorem on Page 158. It is stated next forconvenience.

Theorem 29.7.1 Let M be a closed nonempty subset of a metric space (X ,d) and let f :M→ [a,b] be continuous at every point of M. Then there exists a function, g continuous onall of X which coincides with f on M such that g(X)⊆ [a,b] .

The next topic needed is the concept of an infinitely differentiable partition of unity.This was discussed earlier in Lemma 36.1.6.

Definition 29.7.2 Let C be a set whose elements are subsets of Rp.1 Then C is said to belocally finite if for every x ∈ Rp, there exists an open set, Ux containing x such that Ux hasnonempty intersection with only finitely many sets of C.

The following was proved mostly in Theorem 7.5.5.

Lemma 29.7.3 Let C be a set whose elements are open subsets ofRp 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. One can also assume that{Ui}∞

i=1 is locally finite.

Proof: The first part was proved earlier. Since Rp is separable, it is completely sep-arable with a countable basis of balls called B. For each x ∈ H, let U be a ball from Bhaving diameter no more than 1 which is contained in some set of C. This collection ofballs is countable because B is. Let Hm ≡ B(0,m)∩H \(B(0,m−1)∩H) where H0 ≡ /0.Thus each Hm is compact closed and bounded. Let {Ui}km

i=1 ≡ Um be a finite subset of{Ui}∞

i=1 ≡ U which have nonempty intersection with Hm and whose union includes Hm.Thus ∪∞

k=1Uk is a locally finite cover of H. To see this, if x is any point, consider B(x, 1

4

).

Can it intersect a set of Um for arbitrarily large m? If so, x would need to be within 2of Hm for arbitrarily large m. However, this is not possible because it would require that∥x∥ ≥ m− 3 for infinitely many m. Thus this ball can intersect only finitely many sets of∪∞

k=1Uk.Recall Corollary 11.6.8 and Proposition 11.7.3. What is needed is listed here for con-

venience.

Lemma 29.7.4 Let Ω be a complete separable metric space and suppose µ is a completemeasure defined on a σ algebra which contains the Borel sets of Ω which is finite on balls,the closures of these balls being compact. Then µ must be both inner and outer regular.

1The definition applies with no change to a general topological space in place of Rn.