Step 2. We claim that it is sufficient to show that for every measurablef: Ω→Δ(A), there exists a measurableg: Ω→Asuch that for allE∈ FwithP(E)>0and for all openB⊂A,(11.1)integraldisplayEf(ω)(B)dP(ω) =integraldisplayE1g(ω)∈BdP(ω).Suppose we have such ag∗for the equilibriumf∗, defineE={ω:g∗(ω)negationslash∈Brω}, and suppose thatP(E)>0.For some openB,EB:={ω∈E:Brω∩B=∅}hasP(EB)>0, contradicting (11.1). The existence of suchagfor eachfis the conclusion of the next Lemma.✷Exercise 11.3.20Show that for everyµ∈Δ(A), for every internalE∈ Fand internalf: Ω→Δ(A), thereexists an internalg: Ω→Asuch that for all openB,integraltextEf(ω)(B)dP(ω) =integraltextE1g(ω)∈BdP(ω).[Hint: takeAFto be a dense star-finite subset ofA, considergtaking values inAF.]Lemma 11.3.21 (Loeb Space Lyapunov)For every measurablef: Ω→Δ(A), there exists a measurableg:Ω→Asuch that for allE∈ FwithP(E)>0and for all openB⊂A,(11.2)integraldisplayEf(ω)(B)dP(ω) =integraldisplayE1g(ω)∈BdP(ω).PROOF. LetBτ={Bn:n∈N}be a countable basis for the metric topology onA. Since every openBcanbe expressed as an increasing union of elements ofBτ, by Dominated Convergence, it is sufficient to prove theresult for every openB∈ Bτ. We now appeal to overspill in the following way.Forǫ>0, letEF(ǫ)denote the set of finite internal partitions ofΩwith|E|<ǫ, i.e.max{P(E) :E∈ E}<ǫ.Forr>0, letBτ(r) ={Bn:n≤r}. DefineS={ǫ>0 : (∃E ∈EF(ǫ))(∀E∈σ(E))(∀B∈ Bτ(1/ǫ))(∃g: Ω→A)bracketleftbiggvextendsinglevextendsinglevextendsinglevextendsingleintegraldisplayEf(ω)(B)dP(ω)−integraldisplayE1g(ω)∈BdP(ω)vextendsinglevextendsinglevextendsinglevextendsingle<ǫbracketrightbigg.Sis internal, and, by the previous Exercise, contains arbitrarily smallǫ∈R++. By overspill,Scontains a strictlypositive infinitesimal. Take any internalgassociated with that positive infinitesimal.✷11.4Saturation, Star-Finite Maximization Models and CompactificationWe start with some easy observations about some optimization problems. As easy as they are, they contain thekeys to understanding star-finite maximization models. The statements about the existence of optima for star-finitechoice problems come from Transfer of the existence of optima for finite choice problems.1. Consider the problemmaxx∈[0,1]u(x)whereu(x)x·1[0,1)(x).(a) This does not have a solution because, for everyx <1, there is anx′∈(x,1). For everyǫ >0,x= 1−ǫis anǫ-optimum, as it gives utility withinǫofsupx∈[0,1]u(x).(b) For any star-finiteA⊂∗[0,1], the problemmaxx∈Au(x)does have a solution,x⋄, and ifdH(A,∗[0,1])≃0, then◦u(x⋄) = supx∈[0,1]u(x).2. Consider the problemmaxx∈[0,1]v(x)wherev(x) = 1Q(x).(a) This has infinitely many solutions, anyq∈Q.