Oriented Measures* Raphael Cerf. and. Carlo Mariconda. Submitted by Dorothy Maharam Stone. Received June 15, 1994

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1 Ž JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 197, ARTICLE NO 0062 Oreted Mesures* Rphel Cerf CNRS, Uerste Prs Sud, Mthemtque, Btmet ˆ 25, 9105 Orsy Cedex, Frce d Crlo Mrcod Dprtmeto d Mtemtc pur e pplct, Uerst ` d Pdo, Belzo 7, Pdu, Itly Sumtted y Dorothy Mhrm Stoe Receved Jue 15, 199 A vector mesure Ž,, defed o, 1 s oreted f for ech -tuple of dsjot mesurle sets Ž A 1,, A such tht A 1 A the determt 1Ž A1 1Ž A Ž A Ž A 1 s postve We study the rge R of oreted mesure: R Ž E : E hs dscotuty pots, R Ž E : E hs less th 1 dscotuty pots 1996 Acdemc Press, Ic 1 INTRODUCTION A theorem of Lypuov sttes tht the rge R of o-tomc vector mesure o, R Ž A : A mesurle suset of, * We re deeply grteful to Professor Arrgo Cell for suggestg the tl prolem, for hs useful dvce, d for hs wrm ecourgemet; we lso wsh to th Professors Je-Perre Au d ele Frows, who gve the uthors the opportuty to meet together We wrmly th the referee for sg us to e precse out ucler pot cocerg Thoerem 22 We th the DMI of the ENS for the techcl support durg the preprto of ths pper The secod uthor Ž CM ws prtlly supported y grt of the Cosglo Nzole delle Rcerche Ž Grt X96 $1800 Copyrght 1996 y Acdemc Press, Ic All rghts of reproducto y form reserved

2 926 CERF AND MARICONDA cocdes wth the covex set d :01 ½ 5 owever for gve, 01, the usul proofs sed o covextyextreme pots rgumets ; 5 do ot gve y formto out the exstece of ce set E such tht Ž E d Cosder, for stce, the two-dmesol vector mesure Ž A Ž A, A2AB where B s orel suset of, d deotes the Leesgue mesure For ech set E, the equlty Ž E Ž B mples B E Whe the mesure dmts desty f, l 3 showed tht f for ech vector p the set t, : pfž t 0 Ž where s the usul sclr product s fte Ž respectvely coutle uo of tervls, the there exsts set E whch s fte Žresp coutle uo of tervls I our pper 2, we troduced the stroger oretto codto : we sy tht rel fuctos f,, f verfy codto o tervl, 1 f for ech 1,, the determt f1ž x1 f1ž x2 f1ž x f2ž x1 f2ž x2 f2ž x f Ž x f Ž x f Ž x 1 2 s ot equl to zero wheever the x s, re dstct d ts sg s costt o the -tuples Ž x,, x 1 such tht x1 x2 x We showed tht f mesure dmts desty fucto whose compoets re cotuous d stsfy the oretto codto the the set E my e ult such wy tht ts chrcterstc fucto hs t most dscotuty pots Moreover, f 0 1 there exst two such sets E1 d E2 whose chrcterstc fuctos E d E hve exctly 1 2 dscotuty pots Žoe set s eghourhood of wheres the other s ot Our proofs reled upo the fct tht the mp 2 Ž 1 Ž Ž 1 3,, f x dx f x dx

3 ORIENTED MEASURES 927 s dfferetle d hs vertle Jco wheever 1 We lso showed tht wheever fucto x stsfes xž 0 Ž 2 Ž Ž1 Ž Ž Ž1 x 00 d x 0 1 the the fuctos x,, x, x verfy o eghourhood of 0 We ppled these results to the study of rechle sets of costred g-g solutos d to o-covex prolems of the clculus of vrtos I ths pper, we del wth mesures whch re ot ecessrly solutely cotuous wth respect to the Leesgue mesure Oreted Mesure If A,, A re mesurle sets of, 1, y A A we me tht for ll -tuple Ž x,, x 1 1 of A1 A we hve x x A mesure Ž,, 1 1 s sd to e oreted f for ech -tuple of mesurle sets A 1,, A such tht A1 A the determt 1Ž A1 1Ž A Ž A Ž A 1 s postve I ths more geerl frmewor, we gve ew proof of the results stted 2 We crry out deep study of the rge R of the mesure: for ech pot q of ts teror R there exst exctly two dstct dul sets E 1, E2 whose chrcterstc fuctos hve dscotuty pots such tht Ž E1 q Ž E 2 ; the set R cocdes wth so tht the ove set s ope; the set R s strctly covex; ½ d :01 5 pot Ž E elogs to the oudry R of R f d oly f the chrcterstc fucto of E hs less th 1 dscotuty pots; flly, we gve recursve decomposto of the oudry R

4 928 CERF AND MARICONDA 2 ORIENTED MEASURES Throughout the pper, we wll wor wth tervl, equpped wth the Leesgue -feld L Mesurle wll me mesurle wth respect to ths -feld A eglgle set s mesurle set of Leesgue mesure zero A vector mesure o, s coutly ddtve set fucto defed o the Leesgue -feld wth vlues for some teger Notto If A,, A re mesurle sets of, 1,y A1 A we me tht A 1,, A hve o-zero Leesgue mesure d for ll -tuple Ž x,, x 1 of A1 A we hve x1 x Let Ž,, e vector mesure If elogs to L 1 Ž, 1,we ote Ž d, Ž d Ž,, Ž Ž 1 DEFINITION 21 A vector mesure Ž,, o, 1 s sd to e oreted o, f t s o-tomc d f for ech 1,, d for ech -tuple of mesurle sets A 1,, A such tht A1 A the determt 1Ž A1 1Ž A Ž A Ž A s postve 1 Remr If s oreted, the 1 s postve mesure whch ssgs postve vlues to sets of postve Leesgue mesure I prtculr, f I s o-trvl tervl, the Ž I s o-zero Remr If s oreted d I 1,,I re dsjot o-trvl Ž Ž tervls, the the vectors I,, I form ss of 1 A very mportt fct cocerg oreted mesures s tht ther chrcterstc property crres o from sets to postve fuctos Ž Notto If s fucto ts support s the set supp x : x 0 TEOREM 22 Let Ž,, 1 e oreted mesure If 1,, re -tegrle o-egte fuctos such tht supp 1 supp, the the determt 1Ž 1 1Ž Ž Ž s poste 1

5 ORIENTED MEASURES 929 Let us frst stte preprtory lemm LEMMA 23 Let Ž,, 1 e ector mesure d 1,, e mesurle -tegrle fuctos The determt s equl to d d d d 1 ž / 1 1 Ý Ž1 Ž 1 Ž s Ž s d Ž Ž s,,s Proof of Lemm 23 The detty s ovously true wheever 1,, re chrcterstc fuctos The mootoe clss theorem yelds the result Proof of Theorem 22 We pply the lemm The dom of tegrto of the -fold tegrl s reduced to supp 1 supp We frst prove tht the mesure ˆ Ý Ž Ž1 Ž s postve o the product spce Ž supp, L Ž supp, L 1 equpped wth the product -feld Žwhere L deotes the oe-dmesol Leesgue -feld Notce tht the product -feld L does ot cocde geerl wth the -dmesol Leesgue -feld Že, the completo of the -dmesol Borel -feld Cosder frst the cse of suset of supp 1 supp whch s product set A A Ž where the A s re mesurle 1 Necessrly, ech A s suset of supp If oe of the A s s eglgle, the we hve A A d Ž A A detž A,,Ž A 1 ˆ 1 1 s postve y defto Suppose ow some of the A s re eglgle For ech dex,1, Ž there exsts decresg sequece B m mof o-eglgle mesurle susets of supp hvg empty tersecto Ž ths s cosequece of 1 the fct tht supp s ot eglgle Now for ech m we hve A1 Bm Ž 1 A B whece A B A B m ˆ 1 m m s postve By the cotuty of the mesure, we hve Ž A A lm A B 1 A B ˆ 1 ˆŽ 1 m m m so tht ˆ Ž A A 1 s o-egtve It follows tht ˆ s o-egtve o the oole lger of the fte Ž dsjot uo of product sets: ts uque

6 930 CERF AND MARICONDA exteso to the -feld L geerted y these products s lso oegtve The fucto Ž s,,s Ž s Ž s s postve everywhere o ths set d s mesurle wth respect to the -feld L : thus the tegrl Ž s Ž s dž s,,s 1 1 ˆ 1 s postve Remr If s solutely cotuous wth respect to the Leesgue mesure, the Lypuov theorem yelds ltertve proof of Theorem 22 I fct, 1,, Esupp Ž Ž E Necessrly Ž E s o-zero for ech Ž see remr fter defto 21 d the solute cotuty hypothess o mples tht the E s re ot eglgle It follows tht E E d det Ž,,Ž 1 1 detž E,,Ž E 0 1 We shll deote y the suset Ž x,, x :x x 1 1 DEFINITION 2 The mesure s sd to e loclly oreted f for ech -tuple x of there exsts eghourhood V V1 V of x such tht for -tuple of mesurle sets A1 A stsfyg A1 A V1 V, the determt 1Ž A1 1Ž A Ž A Ž A 1 s postve As curosty, we prove the followg: PROPOSITION 25 A loclly oreted mesure o, s oreted o, Proof Let e loclly oreted mesure The compct set c e Ž covered y fte fmly of ope sets V where V I1 I d ŽI re sutervls of, such wy tht for ech -tuple 1 of mesurle sets A1 A stsfyg A1 A V for some, the determt formed wth the frst compoets of the vectors Ž A,, Ž A 1 s postve Let Ž J l le the fte fmly of the toms of the lger geerted y the sets ŽI,, 1Ž thus the J l s re exctly the sets of the form I for some x, Let us remr tht for ech, : x I

7 ORIENTED MEASURES 931 Ž l 1,,l, the product Jl Jl s coted some product 1 I 0 I 1 0 I fct, J J I I l1 l 1 so tht there exts such tht J J I 0 I 0 0 l1 l 1 s ot empty It follows tht J I 0,, J I 0 l1 1 l d y the very costruc- to of the sets J s we ot J I 0,, J I l l l We deote y ˆ the mesure Ý Ž Let Ž A,, A ˆ Ž1 Ž 1 e -tu ple of mesurle sets such tht A1 A The product A1 A s the dsjot uo of the sets Ž A A Ž J J 1 l1 l Ž Ž whe l 1,,l vres Let ow l 1,,l elog to Ether Ž A A Ž J J s empty Ž 1 l l d thus hs zero ˆ me- 1 sure or t s ot empty d ecessrly, Jl J l Proceedg s 1 the proof of Theorem 22, we show tht ˆ s postve mesure o the set Ž J J whece ŽŽ A A Ž J J l l ˆ 1 l l s o- 1 1 egtve Sce the set A1 A s ot eglgle, t lest oe of these sets s ot eglgle Let Ž A A Ž J J 1 l l e such 1 set It s suset of oe of the V s d, moreover, Ž A J 1 l 1 Ž A J whece ŽŽ A J Ž A J l ˆ 1 l l s postve Thus 1 Ž A A s postve ˆ 1 3 ORIENTED MEASURES WIT DENSITIES Oretto Codto We sy tht rel fuctos f 1,, f verfy codto o tervl, f for ech 1,,, the determt f1ž x1 f1ž x f2ž x1 f2ž x f Ž x f Ž x 1 s postve wheever the x s, re such tht x1 x2 x Remr I our prevous pper 2, we dd ot mpose the sg of the ove determt to e postve Whe delg wth cotuous fuctos, coectedess rgumet shows mmedtely tht the sg s costt o the set I our preset frmewor Ž t the mesure level, we fd t coveet to wor wth ths slghtly more restrctve codto

8 932 CERF AND MARICONDA EXAMPLES For 1, codto sttes tht the fucto f1 s postve; for 2, the fuctos f 1, f2 stsfy f d oly f f1 s postve Ž 1 d f f s strctly cresg The fuctos f t t Ž stsfy codto o Žthe correspodg determts re Vdermode determts 1 PROPOSITION 31 Let f,, f e fuctos L Ž, 1 stsfyg the oretto codto o, Let e the mesure o, whose desty wth respect to the Leesgue mesure s f The the mesure Ž,, s oreted 1 Proof Let A A e mesurle sets of, 1 Sce the determt s multler cotuous form, we c wrte f1 f1 f1ž s1 f1ž s 1 A f2ž s1 f2ž s ds 1ds A1 A f f f s f s A Ž Ž 1 A1 A By codto, the tegrd s postve o A A 1 1 If f,, f re of clss C o, 1, we wll deote ther Wros y WŽ f,, f 1 The followg opertol crtero for the fulflmet of the oretto codto hs ee used 2 1 PROPOSITION 32 Let f,, f C Ž, e such tht 1 t, WŽ f Ž t 0,,WŽ f,, f Ž t The f,, f stsfy the oretto codto o, 1 NOTATIONS AND PRELIMINARY LEMMAS Let us troduce some ottos If u 1,,u re vectors of, ther determt s sometmes deoted y detu,,u 1 Let A e mtrx wth rel coeffcets; y det A or A, we deote ts determt For ech, j 1,,,y Aj we me the Ž 1 Ž 1 mtrx oted y removg the th row d the jth colum from A Surprsgly, the followg smple lgerc trc wll ply essetl role the exstece prt of the proof of theorem 51

9 ORIENTED MEASURES 933 LEMMA 1 Let A Ž j 1, j e mtrx wth rel coeff- cets Let x,, x e such tht 1 1,1 1 1, 1 1 1, x x x 0 2,1x1 2, 1x1 2, x0 x x x 0 1,1 1 1, 1 1 1, If det A 0, the A 1x1 x x A Proof Crmer s rule ppled to the ove system yelds Ž 1 A 1,,1 x x A so tht Ž x x x x Ý 1 A A A A sce Ý Ž 1 A s the developmet of the determt A 1 log the frst row The ext lemms volve strogly the oto of oreted mesure LEMMA 2 Let F d G e two dstct susets of, whch re the uo of l d m dsjot closed terls l m F I, G J 1 d let Ž,, e oreted mesure Assume Ž F Ž G 1 The l m; moreoer f F G the l m 1 Proof j1 Let us frst remr tht the symmetrc dfferece Ž I I Ž J J Ž I J Ž I J j ž / ž / 1 l 1 m j j, j, j s the uo of t most l m o-trvl tervls d tht wheever t lest two tervls hve commo oudry pot the ths umer s

10 93 CERF AND MARICONDA smller th l m 1 Sce the tervls I 1,,Il re dsjot, s well s J 1,, J m, we hve Ž I1 Il Ž J1 Jm Ž I1 J1 Ž I1J1 Ž I1J1 Ž I2 Il Ž J2 J m Now, the set Ž I I Ž J J 2 l 2 m s uo of t most l m 2 dsjot tervls Ether I1 J1 or I1 J1 d I1 J1 s tervl I oth cses, Ž I J Ž I J s the uo of t most two tervls Ž t most oe f I d J hve oudry pot commo 1 1 A strghtforwrd ducto gves the result Sce the sets F d G re dstct, FG s ot empty Let A1 A e the coected compoets of FG For 1,, p p, we hve A Ž A F Ž A G, Ž AF Ž AG A Ž FG Ž FG Ž FG ; the set A eg coected, ether A F G or A G F Put 1 f AFG ½ 1 f A GF so tht the equlty Ž F Ž G c e rewrtte s 11Ž A1 p1ž Ap 0 A A 0 Ž Ž p p Suppose p; the frst p equtos mply tht the determt 1Ž A1 1Ž A p A A Ž Ž p 1 p p vshes, whch cotrdcts the fct tht s oreted The followg ottos wll e used throughout the remder of the pper Nottos 3 We shll deote y the set Ž 1,, :1 To ech -tuple Ž,, 1 elogg to we ssocte the two sets E,, E,, odd eve where y coveto, 0 1

11 ORIENTED MEASURES 935 LEMMA Ž Uqueess Let e -dmesol oreted mesure o, Assume the -tuples Ž,, d Ž,, 1 1 of Ž Ž Ž Ž Ž stsfy E E respectely E E The E E Ž resp E E Proof Assume E, E re dstct d ŽE ŽE Now, two possle cses my occur ccordg to the prty of If 2r, the sets E d E re the uo of t most r tervls Lemm 2 mples r r, whch s surd If 2r 1, the sets E d E re the uo of t most r 1 tervls owever, s commo oudry pot Lemm 2 mples Ž r1 Ž r1 1, whch s surd The dul cse ŽE ŽE c e treted smlrly The followg essetl lemm wll e used repetedly LEMMA 5 Let Ž,, 1 e oreted mesure o the terl, d I I I e 1 suterls of, 0 1 The, ge poste, there exst 1 poste rel umers,, such tht 0 l l Ý Ž l Ž l l0 l 0,, 0 d 1 I 0 Proof Cosder the ler system Ž 0 1 Ž 1 Ž 1 Ž 1 Ž Ž I I 1 I 1 I, where s prmeter The determt of the system s Ž Ž 1 det Ž I,,Ž I The mesure eg oreted, s ot zero Moreover, for ech 0,,1, 2 1 1Ž 0 Ž 1Ž 2 Ž 1Ž I 1 I 1 I 2 1 2Ž 0 Ž 2Ž 2 Ž 2Ž I 1 I 1 I 2 1 Ž I Ž 1 Ž I Ž 1 Ž I Ž 1 1Ž I Ž 1 1Ž I1 1 Ž 1 2Ž I Ž 1 2Ž I1 Ž 1 Ž I 1 Ž 1 Ž I 1,

12 936 CERF AND MARICONDA e, Ž 1 2 Ž 1 det Ž I 0,,Ž I 2,Ž I,,Ž I By Crmer s formul, equls The mesure eg or- eted d hve the sme sg so tht s postve wheever s postve Choosg such tht 0mž,,, / 0 1 we ot Ž 1 -tuple whch solves the prolem 5 MAIN RESULT The sttemet of the m result volves Nottos 3 TEOREM 51 Let e oreted mesure o, d let e mesurle fucto defed o, wth lues 0, 1 There exst two -tuples Ž,, d Ž,, 1 1 such tht Ž E dž E If ddto 0 1, the d stsfyg Ž re uque d erfy 1, 1 Remr The mesure eg o-tomc we do ot cre out oudry pots of tervls d we mght wrte, Ž for the mesure of the tervl Ž, Proof We cosder frst the cse 0 1 d we prove the result y ducto o 1 The mesure eg oreted o,, the mps Ž, d Ž, re cotuous d strctly mootoc o, It follows tht there exst uque rel umers d such tht 1 1 Ž, dž, 1 1 Ž Assume the result s true t r 1 We del oly wth the -tuple : exstece of the -tuple correspodg to t r follows from the fct tht t cocdes wth the -tuple correspodg to 1

13 ORIENTED MEASURES 937 Defe for ech 1,, d for ech -tuple Ž d Ž Ž E The ductve ssumpto yelds the exstece of two Ž 1 -tuples Ž,, d Ž,, such tht , d for ech 1,, 1 Put Ž Ý Ž,,,, Ž, odd Ž Ý Ž,,,, Ž eve S Ž 1,, :1 1, 1,,1 Ž Ž Ž Sce Ž,,, d Ž,,, elog to S, the set S s ot empty We show ow tht ether or Ž Ž Ž,,,,,, Ž Ž,,, Ž,,, Ž The equltes yeld for ech 1,, 1 Ý 01 eve 1 Put for, j 1,, Ž Ý 1 1 d d 0 01 odd Ž Ž j1 j j j j j 1, j x 1, d, A, j1

14 938 CERF AND MARICONDA where f j s eve, j ½ 1 f j s odd Wth these ottos the ove equltes ecome 1,,1 Ý j x j 0 j1 Sce the mesure s oreted the the determt A does ot vsh y Theorem 22 We re thus the posto to pply Lemm 1: A 1,,, Ž x Ž 1 Ž 1 1 Ý j j j1 A Smlrly, f we defe for, j 1,, j j j Ž j j Ž j 1, j j1 x 1, d, A, where we hve f j s odd, j ½ 1 f j s eve, A Ž, 1,,1 Ž Ž 1 A The mesure eg oreted, the determts A d A hve the sme sg, s do A d A It follows tht Ž,,, 1 1 Ž d Ž,,, Ž 1 1 hve opposte sgs At ths stge, we prove tht the set S s the grph of cotuous fucto; ths wll mply tht S s coected Let elog to, We re loog for Ž tuple Ž,, stsfyg for ech 1,, 1 2 Ž, Ž, Ý eve Ž Ž Ý Ž,, eve

15 ORIENTED MEASURES 939 or equvletly Ý ,,1,, 2 eve Ž Ž Suppose frst The ove equtos ecome 1 1 Ý Ž Ý 1 21 eve, Ž Ý Ž 1,,1,, 2 eve eve We put Ž,,, d ˆ Ž,,, If s odd, the E 2, 3 1,, Eˆ 2, 3 1, ; f s eve, the E 2, 3,, Eˆ 2, 3 2, 1 I oth cses, the precedg formule c e rewrtte s Ž Ž ˆ 1,,1 E E ; Lemm mples tht E E ˆ Sce ddto 2 1, the ecessrly 2 2,,1 1, Suppose ow Sce , the Lemm 5 yelds the exstece of rel umers 1,, 0, 12 such tht for ech 1,, 1 The fucto 1 1 Ž 1 1 Ý Ž 1 Ž 1 11, 1, 0 Ý Ž 1 1 1, 1 1, 1 11 odd Ý 1, 1 Ž 1 21 eve

16 90 CERF AND MARICONDA stsfes 0 1o, d for ech 1,, eve d,, Ž Ý Ž We re thus led to fd Ž 1 -tuple Ž,, such tht Ž 2 1 Ž d for ech 1,, 1 2 Ý 2 eve Ž, d, 1 1 Ž or equvletly, f we put,,, 2 1,,1 E d 1 Ž Exstece d uqueess of follow from the ductve ssumpto t r 1 I ddto, sce 0 1 o, 1, we hve We c thus defe mp :, 1 such tht for ll -tuple Ž,, 1 Ž 1,, SŽ 2,, Ž 1 Thus S s the grph of By the cotuty of the mesure, the mps,11, re cotuous so tht the set S s closed; moreover, the fucto tes ts vlues the compct set, 1 It follows tht s cotuous eceforth S s coected As cosequece, the mp, eg cotu- ous o S, reches ll the vlues etwee Ž,,, 1 1 d Ž,,, 1 1 I prtculr, there exsts -tuple S such tht Ž Ž Ths -tuple solves the prolem Sce Ž, 1,,1 Ž d Ž 1,, 1, Ž the 11 so tht 1 2 Uqueess of follows from Lemm Cosder ow the cse 0 1 Let Ž m m e sequece of mesurle fuctos such tht 0 m 1 d m coverges to 1 Ž m L, For ech fucto m there exsts uque -tuple such tht Ž E m m d By compctess, we my ssume tht m coverges to some -tuple of Ž Pssg to the lmt, we ot E Ž

17 ORIENTED MEASURES 91 6 TE RANGE OF AN ORIENTED MEASURE Let e oreted mesure o, We deote y R the rge of, e, R Ž A : A mesurle suset of, LEMMA 61 Let e mesurle fucto o,,01 Suppose there exst o-trl terl I of, d poste rel umer such tht 1 o I The the set 1 ½ d:, L I, I Ž 5 s eghourhood of d Proof Let I1 I e o-trvl sutervls of I The me- Ž Ž sure eg oreted, the vectors I 1,, I form ss of The mp Ý : Ž,, Ž I 1 1 s ler somorphsm d s thus ope Let ½ 5 V Ž,, : mx 1 1 Sce Ž V s eghourhood of the org d s coted the set the cocluso follows ½ 5 1 d:lž I,, I Ž Remr The hypothess 1 mples tht elogs to the teror of R Remr The cocluso of Lemm 61 does ot hold for rtrry vector mesure: cosder, for stce, the -dmesol Leesgue mesure Let : R e the fucto defed y Ž E Ž Ž The teror of s the set 1,, :1 COROLLARY 62 The set Ž s coted R

18 92 CERF AND MARICONDA LEMMA 63 The set Ž cocdes wth the set ½ 5 F d :01 Proof The exstece prt of Theorem 51 mples tht F s coted Ž Coversely, let Ž,, elog to 1 ; pplyg Lemm 5 to, d 12, we ot Ž 1 -tuple Ž,, such tht Put 0 Ý Ž Ž 1 0 0,, 0 d 1, 0 Ý Ž 1 0 eve By costructo, we hve 0 1 d Ž so tht elogs to F Ý,1,1 0 odd Ž d E Ž We hve the followg: TEOREM 6 The rge of cocdes wth R; the mp duces homeomorphsm from oto R d mps oto R Proof The surjectvty of follows drectly from Theorem 51 Ijectvty of the restrcto of of s cosequece of the uqueess prt of Theorem 51 together wth Lemm 63 We clm tht Ž s ope Let elog to Lemm 5 llows s usul to fd pecewse costt fucto such tht 0 1 d Ž Ž Clerly there exst postve d sutervl I of, o whch 1 Put I, 1 V : L Ž I, I Lemm 61 mples tht the set ž / ½ 5 I, I, V d : V Ž I, s eghourhood of Sce ech elemet of V stsfes Ž I, 01, the V s etrely coted F Moreover, F cocdes wth Ž d thus Ž s eghourhood of Ž

19 ORIENTED MEASURES 93 Now ech ope covex set s the teror of ts closure; y Lemm 63, the set Ž s covex d ts closure s R, whece Ž R Flly, we show tht the mp s proper Že, tht the verse mge of compct suset s compct Let K e compct suset of F d Ž m 1 e sequece Ž K such tht Ž m coverges to Ž m for some, 01 Sce the sequece Ž m m s coted,y compctess, we my ssume tht m coverges to By the cotuty of, we hve E d Ž Ž The uqueess prt of Theorem 51 mples tht elogs to The mp s proper d thus closed It follows tht ts verse 1 s cotuous The equlty Ž R s cosequece of the cluso Ž R d the fct tht s oe to oe We refer to 7 for the deftos of clsscl otos ssocted wth covex sets We hve the followg: TEOREM 65 The rge R of oreted mesure s strctly coex Proof Let Ž E, Ž F e two dstct pots of R By Theorem 51, we my ssume tht the sets E d F re fte uos of closed tervls Let 0, 1 d put Ž 1 E F Assume, for stce, E F The there exsts o-trvl tervl I such tht x I Ž x Ž x Ž 1 Ž x E Put mž,1 Lemm 61 ppled to, I, shows tht Ž elogs to R COROLLARY 66 Let E e mesurle suset of, The Ž E elogs to the oudry of R f d oly f there exsts set F whch s fte uo of terls such tht F hs less th 1 dscotuty pots d EFs-eglgle Ž such set hs lso zero Leesgue mesure Proof We frst remr tht the fmly of the sets whch re fte uo of tervls d whose chrcterstc fucto hs less th 1 dscotuty pots cocdes wth the fmly E : E : Ž Theorem 6 shows tht F elogs to R wheever F E or F E for some Coversely let E e such tht Ž E elogs to R Theorem 6 yelds the exstece of -tuple elogg to such tht Ž E Ž E ; cosequece of Theorem 65 s tht Ž E s extreme pot of R The Olech Theorem 5, Th 1 mples tht E E s -eglgle F

20 9 CERF AND MARICONDA Our pproch dscloses the recursve structure of the oudry of the rge of oreted mesure For elogg to 0,, let Ž Ž R E :, R E : Ž Notce tht, R 0, R, PROPOSITION 67 The fucto Ž E R Žresp Ž E R s homeomorphsm from oto ts rge whch cocdes wth R Ž resp R Proof Ijectvty follows drectly from Corollry 66 The rest of the proof uses the techques of the proof of Theorem 6 Remr For ech 1,, 1, the set R R1 s prttoed to two coected compoets R, R owever, for, R R R These results yeld the followg: PROPOSITION 68 The oudry of the rge R of oreted -dmesol mesure dmts the decomposto Ž R R R 0, R R Let T e the symmetry wth respect to Ž, 2 Žso tht for ech mesurle suset A of,, TŽ Ž A Ž, A The for ech elogg to 0,,, we he Ž Ž T R R, T R R REFERENCES 1 R Cerf d C Mrcod, O g-g costred solutos of cotrol system, SIAM J Cotrol Optm 33, No 2 Ž 1995, R Cerf d C Mrcod, Oreted mesures wth cotuous destes d the g-g prcple, J Fuct Al 126, No 2 Ž 199, l, O geerlzto of theorem of Lypuov, J Mth Al Appl 10 Ž 1965, P lmos, The rge of vector mesure, Bull Amer Mth Soc 5 Ž 198, C Olech, The Lypuov theorem: ts extesos d pplctos, Methods of Nocovex Alyss, Lecture Notes Mthemtcs, Vol 16, pp 8103, Sprger- Verlg, BerlNew Yor, W Rud, Rel d Complex Alyss, McGrwll, New Yor, F A Vlete, Covex sets, McGrwll, New Yor, 196