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Introduction
Inmathematics,anonmeasurablesetisoneforwhichthe"volume"can-
notbeassigned.This"volume"canbeunderstooddifferentlydepending
onthestructureinwhichwesearchsuchaset.Infact,fromtheverybe-
ginning,thenotationofnonmeasurablesetwasasourceofconsiderable
controversy.
Thefirstexamplesoftheexistenceofnonmeasurablesetsare:Vitali
set(1905)andBernsteinset(1908).Boththesesetswereconstructedon
thereallineRandunderassumptionofAC.Later,aseriesofexamples
emerged:Sierpińskiset(1938),Lusinset(1956).In1984,Shelahproved
theexistenceofanonmeasurablesetassumingZF+DC.
Interestingly,in1970,Solovayconstructedamodelinwhichallsubsets
ofthereallineRaremeasurable.Indeed,itsresultdependsontheexistence
ofaninaccessiblenumber,theexistenceandconsistencyofwhichcannot
beprovedwithstandardsettheory.Fromaroundthe70softhelastcentury
intensiveresearchwascarriedoutonfurtherproofsoftheexistenceof
nonmeasurablesets.Thisresearchwasfocusedinpartsonsolutionstothe
Kuratowski’sproblemof1935(see[59]).
Astheresultsfromthosepapers,suchanonmeasurablesetmaytake
theformofaunionof"small"sets(inthesenseofthestructureundercon-
sideration).Subsequentstudiesprovidednewexamplesoftheexistenceof
suchunionsofnonmeasurablesetsinvariousstructures(seePartIIofthis
monograph).AsinthecaseofBernsteinset,alsotheexistenceofthese
nonmeasurableunionsiswidelyusedinproofsofoftenwell-knowntheo-
rems-asshowninPartIIIofthismonograph.
Moreover,itisnotnecessarytosearchtheexistenceofnonmeasurable
setsastheunionof"small"setsthatareelementsofapartitionofsetsin
agivenstructure(hencethenotionKuratowskipartition).Theycanalsobe
foundinpoint-finitecoversofsetsbelongingtothesestructures,asshown
inPartIVofthismonograph.
