The kernel associated with a measures H-cone M

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1 The kerel associated with a measures H-coe M D. Mărgiea, "Petru Maior" Uiversity of Tg. Mureş Abstract I this paper we defie a proper kerel associated with a σ-fiite measures H-coe M ad we show that this kerel satisfies the complete maximum priciple (C.M.P) ad if V is a proper kerel associated with M such that V 2 f 0 <, where f 0 pb, 0 < f 0 1 such that V f 0 is bouded the there exists a sub-markovia resolvet V = (V α ) α>0 o (E, B) such that iitial kerel is V. Keywords: H-coe M, kerel associated with M, excessive measure. 1 Itroductio A ordered covex coe S is called H-coe if the followig axioms are satisfied: 1. For ay o-empty family A S there exists A ad we have s + A = (s + A), for ay s S; 2. For ay icreasig ad domiated family A S there exists A ad we have (A + u) = A + u, for ay u S; 3. S satisfies the Riesz decompositio property i.e. for ay s, s 1, s 2 S such that s s 1 + s 2, there exist t 1, t 2 S satisfyig s = t 1 + t 2, t 1 s 1, t 2 s 2. Let (E, B) be a Lusi measurable space ad U = (U α ) α>0 be a proper sub- Markovia resolvet of kerels o (E, B) such that the set of all B-measurable, U-excessive fuctios is mi-stable, cotais the positive costat fuctios ad geerates B. We deote by Exc U the set of all σ-fiite U-excessive measures. It remid that a U-excessive measure o (E, B) is a positive measure ξ o (E, B) provided ξ(αv α f) ξ(f) for all f pb ad α > 0. 1

2 If µ is a positive measure o (E, B) the µ U is a U-excessive measure. A measure ξ Exc U is called potetial provided there exists a measure µ o (E, B) such that ξ = µ U. We kow that for ay family (ξ i ) U-excessive measures ξ i is excessive ad if family is a icreasig ad domiated by a excessive measure the ξ i is a U-excessive. For every σ-fiite measure µ o (E, B) such that there exists ξ Exc U with µ ξ, we put R(µ) = {ξ Exc U ξ µ}. If ξ 1, ξ 2 Exc U ad ξ 2 ξ 1 the exists ξ Exc U such that R(ξ 1 ξ 2 ) + ξ = ξ 1. Therefore Exc U is H-coe ad particularly coe of potetials. 2 The kerel associated with a measures H-coe M Defiitio 1. A proper kerel V o (E, B) is amed associated with M if for ay µ M f +(E) we have µ V M ad follows assertios hold: 1. µ V = ν V µ = ν; 2. ξ M, ξ µ V ( ) ν M f +(E) with ξ = ν V ; 3. ξ M ( )(µ ) M f +(E), with µ V ξ. 4. µ V ν V µ(1) ν(1); 5. If F E is closed i τ topology the for ay µ M f +(E) we have R F (µ V ) = µ V, where µ is carried by F. 6. R [f>0] (µ V )(f) = (µ V )(f), for all µ M f +(E), f pb. Remark. If E is semisaturated with respect to U the U is associated with Exc U. We deote M = {µ M σf + (E) ( )(µ i ) M, icreasig to µ.} Propositio 1. Let M be H-coe of fiite measures o (E, B). The M = {µ M σf + (E) ( ) (µ i ) M icreasig to µ} is also a H-coe of σ-fiite measures o (E, B), such that M is a solid subcoe ad icreasigly dese i M. Proof. Obviously that M is a covex, ordered coe ad for all ν M, ν = (µ ν) 2

3 Let ν M ( ) (ν i ) M icreasig such that ν i ν. The ν µ = ν i µ = i µ) M. (µ Hece for all ν M ad µ M we have ν µ M. It follows that for all µ M ad ν M, such that ν µ ν M. (F 1 ) Let (ν i ) M a icreasig family ad domiated by a elemet of M. Sice ν i = (µ ν i ) (µ ν i ) = (µ ν i ) M. Therefore ν i M. (F 2 ) Let F S a o-empty family. For µ F, we deote by ν µ = (ν µ) ad with ν = ν µ. The family (ν µ ) is icreasig ad domiated by ay ν F. It follows that ν M ad ν ν, for all ν F. Let ν M ad ν ν, for all ν F. It follows that ν µ ν µ, for all µ M, ν µ ν µ. Hece ν µ ν µ, for all µ M. Therefore ν ν. From above it results that ν = ν. (F 3 ) Let ν, ν 1, ν 2 M, such that ν ν 1 + ν 2. For ay µ M, we deote with ν µ 1 = R(µ ν ν 2 µ). It exists ν µ M such that ν µ 1 + ν µ = ν µ. We deote with ν µ 2 = ν µ. We have νµ i ν i, ad the family (ν µ i ) icreasig (i = 1, 2). Let ν i = ν µ i µ µ, (i = 1, 2). The ν i M ad ν i ν i, (i = 1, 2). From ν ν µ ν µ 1 + ν µ 2, for all µ M ν ν 1 + ν 2 ad from ν µ ν µ = ν µ 1 +ν µ ν 1 +ν µ, for all µ µ, we have that ν µ ν 1 +ν µ 2 ν 1 +ν 2, for all µ M. Therefore ν = ν 1 + ν 2. Hece M satisfies the Riesz properties. From (F 1 ), (F 2 ), (F 3 ) it follows that M is a H-coe. Propositio 2. Assume that a proper kerel V is associated with M. The V satisfies the complete maximum priciple. 3

4 Proof. Let f, g bpb, with V f V g + 1 o [f > 0]. We show V f V g + 1. Is sufficietly to show if that µ M f +(E), we have µ(v f) µ(v g) + µ(1). Let F be a icreasig sequece of closed sets with F [f > 0] such that sup(µ V )(F ) = µ V (1 [f>0] ). We deote with ν M f +(E), a measure carried by F such that R F (µ V ) = µ V. We have µ (V f) µ (V g) + µ (1) µ(v g) + µ(1). µ(v f) = R [f>0] (µ V )(f) = R =1 F (µ V )(f) = sup R F (µ V )(f) = sup µ (V f) It follows that µ(v f) µ(v g) + µ(1). Sice µ M f +(E) is arbitrary, we deduce that V f V g + 1. Propositio 3. Assume that proper kerel V associated with M has properties V 2 f 0 <, where f 0 pb, 0 < f 0 1 such that V f 0 bouded. The there exists a sub-markovia resolvet V = (V α ) α>0 o (E, B), havig V as iitial kerel. Proof. Let f 0 be above. Sice V f 0 > 0(ε x V = 0 V if V f 0 (x) = 0 ε x = 0, cotradictio) we deduce that the sequece (A ) of B, such that A = [V f 0 1 ] is icreasig ad A = E. Sice V f A V 2 f 0 1 V (1 A ). From V 2 f 0 < V (1 A ) <, for all N. We have that R E A 0 V f 0 1 if RE A 0 (V f 0 ) = 0. From the theorem Kodo-Taylor-Hirsch it follows that exists a sub-markovia resolvet V = (V α ) α>0 o (E, B), such that iitial kerel is V. Theorem 1. (Kodo-Taylor-Hirsch). Let V be a proper kerel o (E, B) satisfyig the complete maximum priciple. The the followig assertios are equivalet. kerel. 1) There exists a sub-markovia resolvet of kerels o (E, B) havig V as iitial 2) There exist a strictly positive fuctio f 0 pb with V f 0 < ad a icreasig sequece (A ) N i B such that V (1 A ) is bouded for all, A = [V f 0 > 0] ad if RE\A 0 (V f 0 ) = 0. 3) There exist a strictly positive fuctio f 0 pb with V f 0 < ad a icreasig sequece (A ) i B such that V (1 A ) is bouded for all, A = [V f 0 > 0] ad if RE\A 0 (V f 0 ) = 0. 4

5 Propositio 4. Assume that M = M. Let V be a proper kerel associated with M ad V = (V α ) α>0 a sub-markovia resolvet o (E, B), such that V is iitial kerel. The ξ M ξ Exc V. Proof. Let ξ M ad (µ ) M f +(E), such that µ V ξ. Obviously µ V is σ-fiite ad V-supermedia. We deduce that µ V Exc V ad so ξ Exc V. Coversely if ξ Exc V the we have ξ αv α ξ. From V = V α + αv α V ad puttig µ α = α(ξ ξ αv α ), we deduce that µ α M σf + (E) ad µ α V = ξ αv α, i.e. ξ αv α M. O the other had, sice ξ M σf + (E) ad ξ αv α ξ ξ M. Corollary 1. Let ξ M ad assume that there exists a proper kerel V with V 2 f 0 <, where f 0 pb, 0 < f 0 1, such that V f 0 bouded. The for every g pb we have ξ(g) = 0 ξ(v g) = 0. Proof. As i P.3 ξ Exc V. ξ(g) = 0 ξ(v α g) = 0, α > 0 ξ(v g) = 0 ξ(v g) = 0 (αv α g) = 0, α > 0 ξ(g) = 0. Refereces [1] Boboc, N., Bezea L.: Potetial Tehory ad Right Procesees, Dordrecht, Kluwer, [2] Boboc, N., Bucur Gh., Corea A.: Order ad Covexity i Potetial Theory: H-coes, Spriger-Verlag, Berli - Heidelberg - New York, [3] Boboc N., Bucur Gh.: Măsură şi capacitate, Editura ştiițifică şi eciclopedică, Bucureşti, [4] Boboc N., Bucur Gh.: Couri covexe de fucții cotiue pe spații compacte, Editura Academiei R.S.R, [5] Kodo, R.: O Potetials kerels satisfayig the complete maximum priciple, Proc. japa Acad. 44 (1968),