Ultimate L

S=\left<S_\alpha\mid\alpha<λight>beapartitionofS^λ_\omegaintostationarysets.

(WecouldjustaswelluseS^λ_{\le\gamma}foranyfixed\gamma<δ.Recallthat

S^λ_{\le\gamma}=\{\alpha<λ\mid{mcf}(\alpha)\le\gamma\}

andsimilarlyforS^λ_\gamma=S^λ_{=\gamma}andS^λ_{<\gamma}.)

Itisawell-knownresultofSolovaythatsuchpartitionsexist.

Hughactuallygaveaquicksketchofacrazyproofofthisfact:Otherwise，attemptingtoproducesuchapartitionoughttofail，andwecanthereforeobtainaneasilydefinableλ-completeultrafilter{\mathcalV}onλ.Thedefinabilityinfactensuresthat{\mathcalV}\inV^λ/{\mathcalV}，contradiction.Wewillencounterasimilardefinablesplittingargumentinthethirdlecture.

LetXconsistofthosea\inP_δ(λ)suchthat，letting\beta=\sup(a)，wehave{mcf}(\beta)>\omega，and

a=\{\alpha<\beta\midS_\alpha\cap\betaisstationaryin\beta\}.

Thenfis1-1onXsince，bydefinition，anya\inXcanbereconstructedfrom\vecSand\sup(a).AllthatneedsarguingisthatX\inUforanynormalfinemeasureUonP_δ(λ).(ThisshowsthattodefineU-measure1sets，weonlyneedapartition\vecSofS^λ_\omegaintostationarysets.)

Letj:VoMbetheultrapowerembeddinggeneratedbyU，so

U=\{A\inP_δ(λ)\midj‘λ\inj(A)\}.

Weneedtoverifythatj‘λ\inj(X).First，notethatj‘λ\inM.Lettingau=\sup(j‘λ)，wethenhavethatM\models{mcf}(au)=λ.Since

M\modelsj(λ)\geauisregular，

itfollowsthatau<j(λ).Let\left<T_\beta\mid\beta<j(λ)ight>=j(\left<S_\alpha\mid\alpha<λight>).InM，theT_\betapartitionS^{j(λ)}_\omegaintostationarysets.Let

A=\{\beta