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1、StochasticOrderInducedbyaMeasurablePreorderbyDavidC.NachmanDepartmentofFinanceJ.MackRobinsonCollegeofBusinessGeorgiaStateUniversityAtlanta,Georgia30303-3083Phone:404-651-1696Fax:404-651-2630E-mail:dnachman@gsu.eduNovember,2005Abstract.Kamae,et.al.[8,Theorem1]presentsageneralcharacterizatio
2、nofthepartialorderingofprobabilitymeasuresinducedbyaclosedpartialorderingontheunderlyingPolishstatespace.Apreorderisareflexiveandtransitive,butnotnecessarilyantisymetricrelation.Thispaperpresentsasimilarcharacterizationofthepreorderingofprobabilitymeasuresinducedbyameasurablepreorderingont
3、heunderlyingPolishspace.Wethenapplythisresulttoobtainacharacterizationofstochasticmajorization,thepreorderinducedbythewidelyappliedmajorizationpreorderonEuclideanspace.Weshowthatamultifunctionassociatedwiththispreorderiscompactandconvexvaluedandcontinuous,andhencesatisfiesthehypothesesofou
4、rcharacterization.Thecontinuitypropertiesofthemajorizationorderingandtheinducedstochasticmajorizationorderinghavenotbeenwidelyrecognizedandareofinterestintheirownright.Kamae,et.al.[8,Theorem1]presentsageneralcharacterizationofthepartialorderingofprobabilitymeasuresinducedbyaclosedpartialor
5、deringontheunderlyingPolishstatespace.Apreorderisareflexiveandtransitive,butnotnecessarilyantisymetricrelation,andsoisaweakerkindofrelationthanapartialorder.Theintentionhereistopresentasimilarcharacterizationofthepreorderingofprobabilitymeasuresinducedbyameasurablepreorderingontheunderlyin
6、gPolishspace.MarshallandOlkin[9,Ch.17.C]alreadyhintthatKamae,et.al.[8,Proposition1]is“...amuchmoregeneralversioninvolvinggeneralpreorders.”Specifically,weexploitthepropertiesofthepreorderandtheoremsofStrassen[11]andHimmelbergandVanVleck[6]toprovideaKamae,et.al.likecharacterizationoftheindu
7、cedstochasticpreorder.Wethenapplythisresulttoobtainacharacterizationofstochasticmajorization,MarshallandOlkin[9,Ch.11],thepreorderinducedbythewidelyappliedmajorizationpreorderonEuclideanspace.Weshowthatamultifunctionassociatedwiththispreorderiscompactandconvex
