696 CHAPTER 21. THE BOCHNER INTEGRAL
second sort are totally bounded while those functions of the first sort don’t. Assume alwaysthat 1/n < dist
(Gε ,Ω
C). Using Minkowski’s inequality,∥∥∥uXGε
−uXGε∗ψn
∥∥∥Lp(Gε ;W)
=
(∫Rm
∥∥∥∥∫Rm
(uXGε
(x)−uXGε(x−y)
)ψn (y)dy
∥∥∥∥p
Wdx)1/p
≤∫
B(0,1/n)ψn (y)
(∫Rm
∥∥∥(uXGε(x)−uXGε
(x−y))∥∥∥p
Wdx)1/p
dy
≤∫
B(0,1/n)ψn (y)
(∫Rm∥(ũ(x)− ũ(x−y))∥p
W dx)1/p
dy
≤∫
B(0,1/n)ψn (y)
( ∫Rm
η
50(2p−1)M∥ũ(x)− ũ(x−y)∥p
V
+Cη ∥ũ(x)− ũ(x−y)∥pU dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)
( ∫Rm
η
50(2p−1)M2p−12
(∥ũ(x)∥p
V
)dx
+Cη
∫Rm ∥ũ(x)− ũ(x−y)∥p
U dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)( ∫
Rmη
25M
(∥ũ(x)∥p
V
)dx
+∫Rm Cη ∥ũ(x)− ũ(x−y)∥p
U dx
)1/p
dy
≤∫
B(0, 1n )
ψn (y)(
η
25+∫Rm
Cη ∥ũ(x)− ũ(x−y)∥pU dx
)1/p
dy
By assumption 21.10.42, there exists N such that if n≥ N, then |y|< 1n and for all u ∈A ,∥∥∥uXGε
−uXGε∗ψn
∥∥∥Lp(Gε ;W)
≤
∫B(0, 1
n )ψn (y)
(η
25+
η p
8p
)1/p
dy
≤∫
B(0, 1n )
ψn (y)(
η
25+
η
8
)dy
=η
25+
η
8
Recall η < 1.Let n be this large. Then let
{ukXGε
∗ψn
}r
k=1be a η/8 net for Aεn in Lp
(Gε ;W
).
Then consider the balls B(
ukXGε, η
4
)in Lp
(Gε ;W
). If wXGε
is in Aε , is it in some
B(
ukXGε, η
2
)? By what was just shown, there is k such that∥∥∥wXGε
∗ψn−ukXGε∗ψn
∥∥∥Lp(Gε ;W)
<η
8