Step 2 we claim that it is sufficient to show that

Step 2. We claim that it is sufficient to show that for every measurable f : Ω → Δ( A ) , there exists a measurable
g : Ω → A such that for all E ∈ F with P ( E ) > 0 and for all open B ⊂ A ,
(11.1)
integraldisplay
E
f ( ω )( B ) dP ( ω ) =
integraldisplay
E
1
g ( ω ) ∈ B
dP ( ω ) .
Suppose we have such a g
∗
for the equilibrium f
∗
, define E = { ω : g
∗
( ω ) negationslash∈ Br
ω
} , and suppose that P ( E ) > 0 .
For some open B , E
B
:= { ω ∈ E : Br
ω
∩ B = ∅ } has P ( E
B
) > 0 , contradicting (11.1). The existence of such
a g for each f is the conclusion of the next Lemma. ✷
Exercise 11.3.20 Show that for every µ ∈ Δ( A ) , for every internal E ∈ F and internal f : Ω → Δ( A ) , there
exists an internal g : Ω → A such that for all open B ,
integraltext
E
f ( ω )( B ) dP ( ω ) =
integraltext
E
1
g ( ω ) ∈ B
dP ( ω ) . [ Hint: take A
F
to be a dense star-finite subset of A , consider g taking values in A
F
. ]
Lemma 11.3.21 (Loeb Space Lyapunov) For every measurable f : Ω → Δ( A ) , there exists a measurable g :
Ω → A such that for all E ∈ F with P ( E ) > 0 and for all open B ⊂ A ,
(11.2)
integraldisplay
E
f ( ω )( B ) dP ( ω ) =
integraldisplay
E
1
g ( ω ) ∈ B
dP ( ω ) .
P
ROOF
. Let B
τ
= { B
n
: n ∈ N } be a countable basis for the metric topology on A . Since every open B can
be expressed as an increasing union of elements of B
τ
, by Dominated Convergence, it is sufficient to prove the
result for every open B ∈ B
τ
. We now appeal to overspill in the following way.
For ǫ> 0 , let E
F
( ǫ ) denote the set of finite internal partitions of Ω with |E| <ǫ , i.e. max { P ( E ) : E ∈ E} <ǫ .
For r> 0 , let B
τ
( r ) = { B
n
: n ≤ r } . Define
S = { ǫ> 0 : ( ∃E ∈ E
F
( ǫ ))( ∀ E ∈ σ ( E ))( ∀ B ∈ B
τ
(1 /ǫ ))( ∃ g : Ω → A )
bracketleftbiggvextendsingle
vextendsingle
vextendsingle
vextendsingle
integraldisplay
E
f ( ω )( B ) dP ( ω ) −
integraldisplay
E
1
g ( ω ) ∈ B
dP ( ω )
vextendsingle
vextendsingle
vextendsingle
vextendsingle
<ǫ
bracketrightbigg
.
S is internal, and, by the previous Exercise, contains arbitrarily small ǫ ∈ R
++
. By overspill, S contains a strictly
positive infinitesimal. Take any internal g associated with that positive infinitesimal. ✷
11.4 Saturation, Star-Finite Maximization Models and Compactification
We start with some easy observations about some optimization problems. As easy as they are, they contain the
keys to understanding star-finite maximization models. The statements about the existence of optima for star-finite
choice problems come from Transfer of the existence of optima for finite choice problems.
1. Consider the problem max
x ∈ [0 , 1]
u ( x ) where u ( x ) x · 1
[0 , 1)
( x ) .
(a) This does not have a solution because, for every x < 1 , there is an x
′
∈ ( x, 1) . For every ǫ > 0 ,
x = 1 − ǫ is an ǫ -optimum, as it gives utility within ǫ of sup
x ∈ [0 , 1]
u ( x ) .
(b) For any star-finite A ⊂
∗
[0 , 1] , the problem max
x ∈ A
u ( x ) does have a solution, x
⋄
, and if d
H
( A,
∗
[0 , 1]) ≃
0 , then ◦ u ( x ⋄ ) = sup
x ∈ [0 , 1]
u ( x ) .
2. Consider the problem max
x ∈ [0 , 1]
v ( x ) where v ( x ) = 1
Q
( x ) .
(a) This has infinitely many solutions, any q ∈ Q .

11.4. SATURATION, STAR-FINITE MAXIMIZATION MODELS AND COMPACTIFICATION 465
(b) For any star-finite A ⊂
∗
[0 , 1] , the problem max
x ∈ A
v ( x ) does have a solution, x
⋄
. If A contains no
rationals, then even if d
H
( A,
∗
[0 , 1]) ≃ 0 ,
◦
v ( x
⋄
) = 0 < max
x ∈ [0 , 1]
v ( x ) .