This material has already been mostly discussed starting on Page 3336. However, that
was a long time ago and it seems like it might be good to go over it again and so, for the
sake of convenience, here it is again.
Definition 36.1.1Let ℭ be a set whose elements are subsets of ℝ^{n}.^{1}Then ℭ is said to be locally finiteif for every x ∈ ℝ^{n}, there exists an open set, U_{x}containing x such that U_{x}has nonempty intersection with only finitely many setsof ℭ.
Lemma 36.1.2Let ℭ be a set whose elements are open subsets of ℝ^{n}and suppose∪ℭ ⊇ H, a closed set. Then there exists a countable list of open sets,
{Ui}
_{i=1}^{∞}suchthat each U_{i}is bounded, each U_{i}is a subset of some set of ℭ, and ∪_{i=1}^{∞}U_{i}⊇ H.
Proof: Let W_{k}≡ B
(0,k)
,W_{0} = W_{−1} = ∅. For each x ∈ H ∩W_{k} there exists an
open set, U_{x} such that U_{x} is a subset of some set of ℭ and U_{x}⊆ W_{k+1}∖W_{k−1}. Then
since H ∩W_{k} is compact, there exist finitely many of these sets,
{ k}
Ui
_{i=1}^{m(k)
} whose
union contains H ∩W_{k}. If H ∩W_{k} = ∅, let m
(k)
= 0 and there are no such sets
obtained.The desired countable list of open sets is ∪_{k=1}^{∞}
{ k}
Ui
_{i=1}^{m(k)
}. Each open set
in this list is bounded. Furthermore, if x ∈ ℝ^{n}, then x ∈ W_{k} where k is the
first positive integer with x ∈ W_{k}. Then W_{k}∖W_{k−1} is an open set containing
x and this open set can have nonempty intersection only with with a set of
{ k}
Ui
_{i=1}^{m(k)
}∪
{ k− 1}
Ui
_{i=1}^{m(k−1)
}, a finite list of sets. Therefore, ∪_{k=1}^{∞}
{ k}
Ui
_{i=1}^{m(k)
} is
locally finite.
The set,
{Ui}
_{i=1}^{∞} is said to be a locally finite cover of H. The following lemma gives
some important reasons why a locally finite list of sets is so significant. First of
all consider the rational numbers,
The set of rational numbers is definitely not locally finite.
Lemma 36.1.3Let 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 adifferentiable function, ψ_{U}havingspt
(ψU)
⊆ U. Then you can define the following finitesum for each x ∈ ℝ^{n}
∑
f (x ) ≡ {ψU (x ) : x ∈ U ∈ ℭ} .
Furthermore, f is also a differentiablefunction^{2}and
∑
Df (x ) = {D ψU (x) : x ∈ U ∈ ℭ}.
Proof:Let p be a limit point of ∪ℭ and let W be an open set which intersects only
finitely many sets of ℭ. Then p must be a limit point of one of these sets. It follows
p ∈∪
{H- : H ∈ ℭ}
and so ∪ℭ⊆∪
{H- : H ∈ ℭ}
. The inclusion in the other direction is
obvious.
Now consider the second assertion. Letting x ∈ ℝ^{n}, there exists an open set, W
intersecting only finitely many open sets of ℭ, U_{1},U_{2},
⋅⋅⋅
,U_{m}. Then for all
y ∈ W,
∑m
f (y ) = ψUi (y)
i=1
and so the desired result is obvious. It merely says that a finite sum of differentiable
functions is differentiable. Recall the following definition.
Definition 36.1.4Let K be a closed subset of an open set, U. K ≺ f ≺ U if f iscontinuous, has values in
[0,1]
, equals 1 on K, and has compact support containedin U.
Lemma 36.1.5Let U be a bounded open set and let K be a closed subset ofU. Then there exist an open set, W, such that W ⊆W⊆ U and a function,f ∈ C_{c}^{∞}
(U)
such that K ≺ f ≺ U.
Proof: The set, K is compact so is at a positive distance from U^{C}. Let
Since W is compact it is at a positive distance from W_{1}^{C} and so h is a well defined
continuous function which has compact support contained in W_{1}, equals 1 on W, and has
values in
[0,1]
. Now let ϕ_{k} be a mollifier. Letting
---
k− 1 < min (dist(K,W C ),2−1dist(W 1,U C)),
it follows that for such k,the function, h∗ϕ_{k}∈ C_{c}^{∞}
(U)
, has values in
[0,1]
, and equals
1 on K. Let f = h ∗ ϕ_{k}.
The above lemma is used repeatedly in the following.
Lemma 36.1.6Let K be a closed set and let
{Vi}
_{i=1}^{∞}be a locally finite list ofbounded open sets whose union contains K. Then there exist functions, ψ_{i}∈ C_{c}^{∞}
(Vi)
such that for all x ∈ K,
∑∞
1 = ψi(x)
i=1
and the function f
(x)
given by
∑∞
f (x) = ψi(x )
i=1
is in C^{∞}
(ℝn)
.
Proof: Let K_{1} = K ∖∪_{i=2}^{∞}V_{i}. Thus K_{1} is compact because K_{1}⊆ V_{1}.
Let
---
K1 ⊆ W1 ⊆ W 1 ⊆ V1
Thus W_{1},V_{2},
⋅⋅⋅
,V_{n} covers K and W_{1}⊆ V_{1}. Suppose W_{1},
⋅⋅⋅
,W_{r} have been defined
such that W_{i}⊆ V_{i} for each i, and W_{1},
⋅⋅⋅
,W_{r},V_{r+1},
⋅⋅⋅
,V_{n} covers K. Then
let
( ∞ ) ( r )
Kr+1 ≡ K ∖ (∪i=r+2Vi ∪ ∪j=1Wj ).
It follows K_{r+1} is compact because K_{r+1}⊆ V_{r+1}. Let W_{r+1} satisfy
---
Kr+1 ⊆ Wr+1 ⊆ W r+1 ⊆ Vr+1
Continuing this way defines a sequence of open sets,
{Wi}
_{i=1}^{∞} with the property
---
Wi ⊆ Vi,K ⊆ ∪∞i=1Wi.
Note
{Wi }
_{i=1}^{∞} is locally finite because the original list,
{Vi}
_{i=1}^{∞} was locally finite.
Now let U_{i} be open sets which satisfy
W-i ⊆ Ui ⊆ Ui ⊆ Vi.
Similarly,
{Ui}
_{i=1}^{∞} is locally finite.
PICT
Since the set,
{Wi}
_{i=1}^{∞} is locally finite, it follows ∪_{i=1}^{∞}W_{i} = ∪_{i=1}^{∞}W_{i} and so it
is possible to define ϕ_{i} and γ, infinitely differentiable functions having compact support
such that
∪_{i=1}^{∞}U_{i} because ϕ_{i} equals one on U_{i}.
Consequently γ
(y)
= 0 for all y near x thanks to the fact that ∪_{i=1}^{∞}U_{i} is
closed and so ψ_{i}(y) = 0 for all y near x. Hence ψ_{i} is infinitely differentiable at
such x. If ∑_{j=1}^{∞}ϕ_{j}(x)≠0, this situation persists near x because each ϕ_{j} is
continuous and so ψ_{i} is infinitely differentiable at such points also thanks to Lemma
36.1.3. Therefore ψ_{i} is infinitely differentiable. If x ∈ K, then γ
(x)
= 1 and so
∑_{j=1}^{∞}ψ_{j}(x) = 1. Clearly 0 ≤ ψ_{i}
(x)
≤ 1 and spt(ψ_{j}) ⊆ V_{j}. This proves the
theorem.
The method of proof of this lemma easily implies the following useful corollary.
Corollary 36.1.7If H is a compact subset of V_{i}for some V_{i}there exists apartition of unity such that ψ_{i}
(x)
= 1 for all x ∈ H in addition to the conclusionof Lemma 36.1.6.
Proof:Keep V_{i} the same but replace V_{j} with
^Vj
≡ V_{j}∖H. Now in the proof above,
applied to this modified collection of open sets, if j≠i,ϕ_{j}
(x )
= 0 whenever x ∈ H.
Therefore, ψ_{i}
(x )
= 1 on H.
Theorem 36.1.8Let H be any closed set and let ℭ be any open cover of H.Then there exist functions
{ψi}
_{i=1}^{∞}such thatspt
(ψi)
is contained in some setof ℭ and ψ_{i}is infinitely differentiable having values in
[0,1]
such that on H,∑_{i=1}^{∞}ψ_{i}
(x)
= 1. Furthermore, the function, f
(x )
≡∑_{i=1}^{∞}ψ_{i}
(x)
is infinitelydifferentiable on ℝ^{n}. Also,spt
(ψi)
⊆ U_{i}where U_{i}is a bounded open set with theproperty that
{Ui}
_{i=1}^{∞}is locally finite and each U_{i}is contained in some set of ℭ.
Proof:By Lemma 36.1.2 there exists an open cover of H composed of bounded
open sets, U_{i} such that each U_{i} is a subset of some set of ℭ and the collection,
{Ui}
_{i=1}^{∞} is locally finite. Then the result follows from Lemma 36.1.6 and Lemma
36.1.3.
Corollary 36.1.9Let H be any closed set and let
{Vi}
_{i=1}^{m}be a finite opencover of H. Then there exist functions
{ϕi}
_{i=1}^{m}such thatspt
(ϕi)
⊆ V_{i}and ϕ_{i}isinfinitely differentiable having values in
[0,1]
such that on H,∑_{i=1}^{m}ϕ_{i}
(x )
= 1.
Proof:By Theorem 36.1.8 there exists a set of functions,
{ψi}
_{i=1}^{∞} having the
properties listed in this theorem relative to the open covering,
{Vi}
_{i=1}^{m}. Let ϕ_{1}
(x )
equal the sum of all ψ_{j}
(x )
such that spt
(ψj)
⊆ V_{1}. Next let ϕ_{2}
(x )
equal the sum of all
ψ_{j}
(x )
which have not already been included and for which spt
(ψj)
⊆ V_{2}. Continue in
this manner. Since the open sets,
{Ui}
_{i=1}^{∞} mentioned in Theorem 36.1.8 are locally
finite, it follows from Lemma 36.1.3 that each ϕ_{i} is infinitely differentiable having support
in V_{i}. This proves the corollary.