# Step 2 we claim that it is sufficient to show that

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Step 2. We claim that it is sufficient to show that for every measurablef: ΩΔ(A), there exists a measurableg: ΩAsuch that for allE∈ FwithP(E)>0and for all openBA,(11.1)integraldisplayEf(ω)(B)dP(ω) =integraldisplayE1g(ω)BdP(ω).Suppose we have such agfor the equilibriumf, defineE={ω:g(ω)negationslash∈Brω}, and suppose thatP(E)>0.For some openB,EB:={ωE:BrωB=}hasP(EB)>0, contradicting (11.1). The existence of suchagfor eachfis the conclusion of the next Lemma.Exercise 11.3.20Show that for everyµΔ(A), for every internalE∈ Fand internalf: ΩΔ(A), thereexists an internalg: ΩAsuch 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: ΩΔ(A), there exists a measurableg:ΩAsuch that for allE∈ FwithP(E)>0and for all openBA,(11.2)integraldisplayEf(ω)(B)dP(ω) =integraldisplayE1g(ω)BdP(ω).PROOF. LetBτ={Bn:nN}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|, i.e.max{P(E) :E∈ E}.Forr>0, letBτ(r) ={Bn:nr}. DefineS={ǫ>0 : (∃E ∈EF(ǫ))(Eσ(E))(B∈ Bτ(1))(g: ΩA)bracketleftbiggvextendsinglevextendsinglevextendsinglevextendsingleintegraldisplayEf(ω)(B)dP(ω)integraldisplayE1g(ω)BdP(ω)vextendsinglevextendsinglevextendsinglevextendsinglebracketrightbigg.Sis internal, and, by the previous Exercise, contains arbitrarily smallǫR++. 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= 1ǫis anǫ-optimum, as it gives utility withinǫofsupx[0,1]u(x).(b) For any star-finiteA[0,1], the problemmaxxAu(x)does have a solution,x, and ifdH(A,[0,1])0, thenu(x) = supx[0,1]u(x).2. Consider the problemmaxx[0,1]v(x)wherev(x) = 1Q(x).(a) This has infinitely many solutions, anyqQ.
11.4. SATURATION, STAR-FINITE MAXIMIZATION MODELS AND COMPACTIFICATION465(b) For any star-finiteA[0,1], the problemmaxxAv(x)does have a solution,x. IfAcontains norationals, then even ifdH(A,[0,1])0,v(x) = 0<maxx[0,1]v(x).

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Term
Fall
Professor
FACKLER
Tags
Metric space, Hilbert space, measure, Compact space, Banach space