Čihák, Pavel

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaePavel ČihákComparability and conditional maximality of measures supported by finite setsof real numbersCommentationes Mathematicae Universitatis Carolinae, Vol. 10 (1969), No. 3, 493--507Persistent URL: http://dml.cz/dmlcz/105247Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1969Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCommentationes Mathematicae Universitatis Carolinae 1 0,3 (1969) COMPARABILITY AND CONDITIONAL MAXIMALITY OF MEASURES SUP-PORTED BY FINITE SETS OF REAL NUMBERS Pavel CIHKK, Praha The notion of ordering of measures, generally intro-duced by G. Choquet, is in the present paper investigated in a relation to stochastic matrices of the type (m>, /*&)• The purpose of this paper is to obtain effective necessa-ry and sufficient conditions for the comparability of mea-sures. These results are applied to finding of conditional-ly maximal measures. Illustration of the present theory is given through some numerical examples. Contents 1. Notations 2. Ordering of a set of measures and stochastic mat-rices 3. Conditions for comparability of measures 4* Applications to conditional maximality of measu-res 5. Numerical examples 1. Notations. Let R ^ be the euclidean space of di-mension tn1 m= 4,2,.., . We shall denote by $r* ay scalar product of the elements Jtr and ay and by c<r{Jb-l convex - 493 -hull of the aet {*% ? Jtr±,... , Jb^ ? f o r any Jlr** (^%?^ € R<» and <%= (tyit)iuHe ^ ' ^ ^ s e t o f a 1 1 conv9X ^aaetiona on the apace R^ wi l l be denoted by C . Define Let fil-r CO.* )• ^ be a matrix of the type (/rn,f<n) . We de-note by Q* = (a„•-)*. . the adjoint matrix of the type 0* , ,m>. If fc^* 0 and^Z ^ « 4 ror £ « 4, . £ , _ , * * and A = /̂̂  2 ; '^ ; <n then the matrix (St i s called atochaatic. If moreover Jlr e R^ and o t e S , then we denote by a, s *Q>Xr and /3 » fl*a auch el omenta a e l ^ . / 3 e -P that d£ wi l l denote a translated Dirac measure on R^ for 2. Ordering of a aet of measures and atochaatic matrieea. (2.1) Let (P be the aet of a l l probability-measures, supported by f i n i t e se ts of real numbers, i . e . where ^rf)==* 21 /^^r i j^ ) for each function f on Ri . (2.2) Let -> be a relation auch that X4(JL i f f A7(u.e (P and ac«P)^ (C^6P> f or all-P <s £ , Then thia relation ia transit ive and ref lexive , i . e . -4 ia a qua si-ordering on the aet (P . The following theorem (2.3) follows from the theorem 2 - 494 -in £2] or from a theorem in -6],chap.13 , but the present proof i s more simple. 12.3) Theorem. Let X <tA, be two elements of the set (P of the form Then X -< (u, i f and only i f there i s a stochastic mat-rix CI of the type (nm^m,) such that Qir-r a, and &*ot =r /S . groof. f Let $ « (%^.^) be a stochastic matrix of the type (an^/n ) Qir-=r a, ? Q*<-£ * /3 . Then *-*> = . ̂ * . ^ ^ ) - Z a y - f t r ^ ^ A : )«£..£. oc, 2 « •- -f C& ) -% £ < £ * * * V f C V ^ a l l * e <*, i . e . A. -«5 <o- . 2* Let X V (U, . Then KQ,^ JlT c c^r-fir* . If the op-posite statement holds f take a convex function -P such that «f -w 0 on the set c<r{j&'?7 «P posi t ive on R —ar{trh Then jex-C-O » 0-cJlC-f) which i s a contradiction. Hence each set S*« ^ - C f ^ ^ e £ / j f / f l i t ^ - V > £ - 4 , 2 , - . , , * * , i s nonempty and convex. Define ^ L (0, 0,...,4x, 0,...,O)e fL,, S*.U «€**>> x S * ) . Hence 0 * S c R _ _ , convex hull cxrCS)-=f= j0f . If (oc; /3) £ c<r CS ) then by C1J there are jp** tyiiX^j* S* *w\, * «%, such t h a t ^ c c ^ * - (3 , i . e . g ^ a* 0 , ^ ^ ^ 4 , & *»*-*&--W^fc**"* " / S ' * ' w - 1 , V „ m i and in. * «tv such t h a t ^ c C ^ * «- ^ , i . e . %M a* 0 } ^ %^= 4 , J '** ^ s ^ 2 . , M , /n , Hence the matrix (k-(%Ajit ) ^8 s t 0 ~ - 495 -chast ic , flir = CL and Q>*cc = (h . If Cot.,/S)e) ior(S) then by Caratheodory'a theorem 151 and Hahn-Banach's theorem 151 there i s a hyperplane that s t r i c t l y separates those s e t s . Since.X<^4^E/^»'f-^Zf^«2 for a l l i^'^P^Jk^6'^^ y there i s a l inear functional FCoC? £ ) > 0 and F-£ 0 on the set S , i . e . there i s an element .* € R^ and an element n^e R^ such that o& * x-/5'tz./, > 0 and x--<f2,*/2^-£ # for a l l ^<^ S ^ Define n^(cL^) ** imfi^-n^; f* e h*l . Then x^ -£ <y,(ag). Let ^ be the greatest convex function on the set ccritrl such that ^ C - ^ > - 4 ^ for jfe « > f , . 2 r . ^ ^ . l f ^>e i1,!7.„fml thea /i^Ca^) - ^ (3u ) , since the point (CL^ , ^ Ca^ )) i s an element of the l ine segment, joining points (Jr^ ,'&£.) and (S/jk 1 style, ) such that Sfa -£ CL^, -£-% - Hence there e-xista 4*>* «>70,...0,^,0,... 0fffk^90,...901cS*', so that 4 C e ^ ) ^ ^ ^ + ^ ^ - ^ ^ £ % > + ^ f c ^ c % * > ~ # ^ > • The convex function ^ has a convex extension in (5 , Using the condition X A (U , we obtain following inequal i t ies rttv an, *& i . e . a contradiction. Now we shall prove that the quasi-ordering on (P i s moidover an ordering. This result i s obtained by theo-rem (2.3) without the c lass i ca l Stone-Weierstrass's theo-rem and i t can be generalised for any linear space R * (2.4) A convex function (yet i s cal led s t r i c t l y convex - 496 -i f f the following condition holds: i f Jfy,iraeR„ Xr^ ir^ te <0, O , <%(ttr^tt-t)irx) = t ^ C ^ ) + - ( 1 - t ) ^ ) t h e n t - 0 or t*4. (2.5) Lemma. Let B he a f i n i t e subset of the set R̂  . Let fi be a nonnegative function *on the set B such that K > f i (ir)*7 Jlr € b}*^. Let ^ be a s t r i c t l y con-vex function in & and l e t Cj.CE Kft Cir)ir-Jtre B } ) ~ Ziii(J^)^(J^)}ire Bl , Then there i s an element ir0 e B such that fiOr0)=4 a n d ^ i C ^ ) » 0 for a l l Sire B7 ^ * ^ . Proof. If iq e B7 j(L(fr^)e(0>1) then there i s an e-lement Sr^e E>7 J2q # fr± , that also ^Cb^) e> COyi) . Put frC£r )+ft(tr2)= e . We obtain the following inequal i t ies: ^ C Z f f t C ^ ) ^ ; ^ ^ + C1~£)<^CC1-£)^Z{^C^)^; ire B, i r - f c i r + i r , 3 ) ^ ^ ^ C i K ) g . C ^ ) - H ^ < : 4 > 9 - c ^ > ^ -^H-£:)c^CC-l-er' fZffiC£r;ir ) ire B, fy^fr* <*%}) ^ ±?-ify(Qr)c%i.ir); ireB}* q,(5iifiCJir)£r; Are & } ) . Hence ^(^fi(fy )fri + £->iC^)i£ >- E'fiC^^Cflr)+ £^C-%)£C%>. Since the function cj- i s s t r i c t l y convex, i t follows that fb(ify)* 0 or fi(£r2)*> 0} which i s a contradiction. Hence fi(ir)=0 or fittr)*? 4 for a l l ir-€ B . Since E{ftC-fr);ir6 &5~ 4, there i s an element Jb; e B such that fi(irc)~iy ftOr) ** 0 for a l l >fre B , ^ 4 = ^ . (2.6) Remark. If QJL i s a measure, f t € ! P , t h e n there i s one and only one expression of (U such that • 497 -(2.7) Theorem. I f A e P. j t t 6 ^ a ^ ^ ? <ix, -$ X then a « ^ . Proof. Let <&> be a function of (5 , which i s s t r i c t -ly convex. Then using the expression X -=.5Loc. <3F r ^ - . 5 1 / 3 A , O £ , oce CP+ fie <P+ CL,>CL >...>&, fy >&- > - - - > ^ and choosing by theorem (2.3) a stochastic matrix Q-fy^fc) of the typeCim^/Tt) such that 0>$r= a> and Q*<x -=- /3 7 we obtain •wt' /n»v /•v ^ ^ ^ c c ^ ^ ) - ^ ^ ^ ^ ^ ^ ^ ) * %£ «*&*** * ' V v * A-9-«fc>- <"<§•> * * c^) • Since <x̂  > 0 an&jLy^^-qrCJLfy^)* (9, i t follows from the equal i ty^* . , C ^ ^ ^ C ^ ) - ^ ^ ^ ^ ) ) = 0 that Lemma (2.5) implies the existence of numbers A C ^ ) e €. -C 4., £.,,.-, /w. ? such that a**^- *> I**, ~ ° for a11 *-**-&>, £ - *,*>->"»- • Prom the fact that Sir and <a.-» <3-#- are both s t r i c t l y de-creasing f i n i t e sequences follows that jfeCD < Mi%X...<MOnt) BX&OYV £ at. i f wt <: m ,then thare i s an integer Jfee <1,2, . . , , , /n.1 such that g ^ — i s a contradiction. Hence ^»/n;JkC/0-r 4; Jk(2}**2.r..,Jk(mi)=: m,, - 498 -flt. i s the unit matrix a - = i r , oC=r/37 A - <«* -^2.8) Corollary. The quasi-ordering -£ is moreover an ordering on the set (P # 3. Conditions for comparability of measures (3.1) Lemma. Suppose ^ , i \ 7 Ufa are real numbers for A,m4,!,„.,*,, %>*,*.„*££, fa* 0, JL^- * and >u.|L £ -ft̂  for all f̂e . Let /*> be such integer of the .6-4 >̂ o set <>V2.,,,,,/n,} that SL tu, ^ 4 - ^ Z ^ (put-Si » 0 ) . Then Proof. P u f 2 £ - - ^ ~ , ^ • ^ i f "̂ fc,*8 ^-Clearly, °*Cv£ * ^ £ } CJ8^~ ^ f0r "•* * ^~ * " We obtain the following inequalities = fl21^- i l + ( 1 - 21 ^c- )>&: . (3.2) Theorem. Suppose X 7 (u, e, (P7 & -4 (*** 7 x=&«i<i> ( " - ^ / M * *-<**£>& > Then the following conditions (for comparability) are ful-filled: - 499 -(3-3) £**%~&fi±*h> and i f / t e ^ V " ^ ^ ^ ^ 1 6 ^ ^ ' " ? ^ ^ Z AA2L**- ^«2l PM. then {3.4> ^5-<. / -v- 4 J i£^^+ iS*t i -4S . ,v^r t> • Proof. Theorem (2.3) impliea there ex i s t s a stochas-t i c matrix Q-rGj-^) of the type (m^ai) euch that QflrssQ, and Q*oC s /& • Using the lemma (3.1) we obtain the f o l l o -wing inequal i t ies: /nv zwt, *TV /rv AIV ^2r (3.5) Theorem. Suppose .A7 Ô/ £ JP . m v At/ a^ *> a^ > ... -̂  ̂ j ^ . Let the conditions for comparabi-lity (3.3) and (3.4) hold. Then X *4 (M* • Esse*. PutK--t*6<r;.gcc^r^) *J^Ab*f-V* . Clearly, K is a convex wedge. From (3*3) follows thac the wedge K contains all affine functions of the set £. We shall prove that K contains also all convex functions <^x for * € R where ^ C | ) « §~X for §> *x, (j^Cf )-*0 for £ £ .x . - 500 -ř-It KeiA927...9an-Jli,xeZii % > * * &>*, + ! then using (3.3) and (3.4) we obtain t h* following inequalities /rtv /c/ M/ H, AfcM -^l^>t-W*)*.5.iv^^~*)^/^*f^)to^^,X w & ** <**> -*£/%/V * ) + ^ 0*-£**>(*-%*>> * -^Z / W - * > -*? , /%>«* «fc.) for .x < ^ , • I f * * % then X <** C-V } " ° *&, &*&* <•**- > ' 1H' /?*v I f ( ^ > ^ then .21 <*,. a/« (-<Z>J ) - .ZLoC-6-tv - -X ) -» /Wt> IV / f t . /TV Now, l e t i be any element of the set 6 -The set of a l l o^ a n d J ^ ? £ e <4,2,...,arni, M,€i492,...,m, } , can be ordered. Suppose <^ < c£J?< ... ^^ are a l l i t s e l e -ments. Define a function to>i JhCcC*)^ f(d* ) for i ^ - V ' v f c and A as an affine function between the numbers c ^ , ^ This function ii/ i s convex, moreover .^ i s a l i n e a r non-negative combination of an affine function and the func-t ions Cfa 9 i s 1, Z>7f7 & - Hence Jh,& K and 4 e K , i . e . K « <*; A, -2 ^ • (3.6) Corollary. Suppose A , (U. e (P, a ^.ZELoc- c£. •Gj i Jlrx ... ± \ . Then the following statements are equivalent : - 501 --> ' 1* X~t <"s > 2° the conditions (3.3) and (3.4) are fulfilled. 4. Applications to conditional maximality of measures. In this section it will be shown how the theorems (3.2) and (3*5) about the comparability of measures can be applied to finding of measures & which are "condition-ally maximal". In particular: /TV (4.1) Theorem. Suppose (U, e (P, (U, »^21 /% cfL 7 *;>.%>... z^p-C&Cetf, *-fi*£«tr . Then there exists one and only one measure 6" e (P, fulfilling the following conditions; 2° C -4 (tt. , 3° if .xe ̂  a-Jla^, a^ a,z> ... > a,^ and 3. -J («, then a,. -* 6" . Proof. Define the following elements of the set $^: -V^T It'n&'a,-;'*'*, °> 0, ... ,0) , where*-** -/,f,...,<*fc and A* — ,-Z oc • , and the following elements of the s e t to $x4 * * ° v-^,i^,-^^.f, V ^ fk, 0, 0,...,0) , where H, =* 47 2,...7 ML- 4 and A>Ot) e <17 2,-*., mi,^fi^^ *oo Then the conditions of comparability (3*3) and (3*4) can - 502 -be written in a simple form (3.3) ir""-. Cb= <ur'm'. Jlr , (3.4) v*. a, & ntr*. 2r i o r ft - 4,2.,*.., /nt-l . The element c ^ f c ^ ^ e R ^ 7 destinating the measure 6* in (4.1), can be chosen as a solution of the following system of sm, independent linear equations nr*. e - <wH'. Jtr for K » i7 2, ...7 mv , i.e. 21 nr:* <L- »^*£jbL • Define a matrix V* Cnr% \ • of the type ftm^/m,) and a matrix W» (mr^ )„ - of the type f !w, /.** ) -.ft fc,.fv 7 Then there ex is ts the inverse matrix V~ and a matrix Clearly, E « , * - * - e , E * , V * - V * E * f / f * * - «ir* for K x j , 17... 7 mv . Now, we shall find a l l elements -e^ of the matrix E - f , . P u t e ^ - ^ O , . . . , ^ ^ 0 ) e t , K-0, v°=0, <vr° -= 07 /*(«*,} a /u, . Then I t follows that <«.-*c«- 4f\r*-v*«^*" f ) -1 /* *<&-f Hence - e ^ - * 0 f o r Jks <: ^>C>t-4 ) . f If bGc-D-kC*,) then € **<W " ^ ť W í ) s *, ^ 4 = ° for A >AC«,) . - 503 -2* I f * (K.-4X/*(*>) then w^«4</w*->W;& /v-4<&/%--**-,>»*, "***• * « **•"£ ^ ^ f o r 'k-*^* -D , •%*-• ° for A > /&^, 0 3 *Ok-'f) >/*>Ot) i s impossible; i . e . a l l elements 4f4t, of the matrix E^ ^ are nonnegative, * e e ^ * 0 for Jtv <<. A>(tt-1) ? k, > 1 and • e ^ ^ =• 0 for A > /t> (jts\fo<<ni.Moreover, s in-c# or""' » *, E* ^ = * r ^ . « / 3 and tor tt~ i7tj...7^7 <>= (i7A,...74)£K„, , we obtain the following statement: The matrix E^. - is stochastic, E ^ m Jlr =? c . «, E / » * - ß I t follows from the theorem (2.3) that 6* ~i pu • On the other hand i f a, ^ art-£..• ^ P ^ - A*.2-*L- <£. , X -*? t̂x then, by theorem (3.2) and definit ion of c , if •Q,*itr .Jlr^ir • c and v a,£w*• xr** nr * c for .1 m /\7l7.t.7 /m . Hence, the conditions for comparability of measures X and ff hold* It follows from the theorem (3.5) that X A <o -It remains to show that c„ --? C„ £: ,„. -̂  c . Clearlyf A>(1) ^A>(2) £ *£ vbf/nt) . Using the proper-ties of -Ciju ,we obtain the following inequalities: - 504 -Hence ̂ * ^ > ^ ±c^ Z £>„ for K, = 4fl,...,m.-4 . If 0" is a measure having also the properties 1° , 2° and 2° in (4.1), then & A &' ^tf'-2, 6" « Hence by the theo-rem (2*7) 6" » 6* . The proof contents moreover the following statement: (4.2) Corollary. Let ot e ̂  and /S e Jg*" be given. Then there exists one and only one stochastic matrix E^ ^ of the type (/rn7m,) such that 1° E * _ at m fi . •rrv c * £ > r - .Z oo. oT then ^ is a greatest element of the set (4.3) U e t P ; A ^ Z ^ - ^ , « ^ a ^ . . . *<%., A -? <«.? . This matrix £,. _ has the following property: (4.4) If AC*)€ U,2,...,aii, ^Z /% - - ^ - ^ %?« /*, for * = 4,.?,...,/»n-'/, /*f/»n>-/n.,^f*^^... > ^ , ' c * » £^-.>, U.5) Nuia.if *-^ f i , . . . , i )«qsr, /s-t'i.i,-. '•'-m, ) £ ^ then the matrix E^ ~ coincides with the doubly-stochastic unit matrix E of the type (nn7nv) introduced in 131 as an exposed element* 505 5 > Numerical examples* (5.1) Suppose (I m Cjt,4-t,ji), oc» (z,,4-z) where te(071),z<i(0,±>, +-(1,0,-4), fL-jt,A>4± • We want to calculate real numbers c. and c. such that the measure &-aid£ + f'f-.£)<£ i s a greatest ele-1 z ment of the set (4.3) 1° If Z,<£t t h « o * 0 ) H . V - M , 0, Q ), f 1 , 0 , 0 ' A«)-3,«rlfit,f-t,|t), *~\^M'XH* 2° I f £ t £ a c < £ t + ( 4 - t ) then-Jt=£a>, \t^i-Z, , oco-^vfi-c-kv-tjt), **i o,i-jfcatj§=a) C = ^'""IfFF)^ ' 3° If j t + (*.-*) ^ .* then 1-« ^ -J t , A M ) « 3 <ur« f-t- - - - - - </-£ + £ - ) T-t i=i i-2z.-2} y.C0«3, <iir-c*ft, *-t, j t ) , ^ " I t f , 0, 4 J In part icular , i f JK - | - then ct >- C j , | ) , i f moreover ^ « J- C4J'+ ^ +- T̂ ) then e-- | « * | + c^ ) . - 506 -(5.2) In order to illustrate the property (4.4) of the conditionally maximal measure & take the followingex-ample: Then A>C1)=-4, MC2)= 2, AC3)=2, /»W= 3, A>(5) = 3, E « , - > - c - 0,2,0,-2,-3). Clearly R e f e r e n c e s CI] N. BOURBAKI: Espaces vectoriels topologiques, Paris. [2J p. CARTIER, J. FELL, P.A. MEYER: Comparaison des mea-sures portées par un ensemble convexe com-pact, Bull.Soc.Math.France 92(1964),435-445. [3] P. 6IHÀK: On an exposed element of a set of doubly-stochastic rectangular matrices, to appear in Comment .Math .Univ • Carolines • [41 G. CHOQUET, P.A. MEYER: Existence et unicité des re-presentations intégrais dans les convexes compacts quelconques, Ann.Inst.Fourier(Gre-noble) 13(1963),139-154. [53 F.A. VALENTINE: Convex sets, New York 1964,15-25» [6] R.R. PHELPS: Lectures on Choquet's theorem, Van No-strand.New York,1966. Matematicko-fysikélnl fakulta KU Sokolovské 83, Praha 8 Ceskoslovensko (Oblatum 5*8.1969) - 507 -

Comparability and conditional maximality of measures supported by finite sets of real numbers

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Comparability and conditional maximality of measures supported by finite sets of real numbers

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