A Result on Diffuse Random Measure

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1 It. J. Cotemp. Math. Si., Vol. 2, 2007, o. 14, Result o Diffuse Radom Measure. Varsei ad. Samimi Departmet of Mathematis Faulty of Siees, The Uiversity of Guila P.O. Box 1914 P.C , Rasht, Ira varsei@guila.a.ir, samimi@guila.a.ir bstrat We prove a result for a diffuse radom measure with Rado itesity, from whih it follows that if φ ad ψ are two radom measures i Γ defied o (M (d), M (d),p) suh that E( φ(i) M (d) I )= E( ψ(i) M(d) I )P a.s. for all I i the uio of a system of a partitios of Γ, the φ = ψ P- a.s. Keywords: radom measure, diffuse, itesity, oditioal, Rado 1 Itrodutio Let Γ be a loally ompat seod outable ausdorff topologial spae, i.e. a polish spae, B the σ -algebra of Borel subsets of Γ, B 0 the family of bouded sets i B ad let M be the spae of o egative Borel measures whih are fiite o B 0, the spae of Rado Measures, ad M be the σ - algebra i M geerated by the mappigs μ μ(b),b B 0, (i.e. smallest σ - algebra makig these mappigs measurable) ad M (d) deotes the subset of M osistig of all diffuse measures ad M (d) = M M (d). For I B, M I deotes the σ - algebra i M geerated by mappigs μ μ(b),b B 0, B I. M (d) I is defied similarly for ay I B. radom measure i Γ is a probability measure o (M, M) or a measurable futio from some probability spae ito (M, M). Diffuse radom measure is defied similarly. If ξ is a radom measure, it is lear that for fixed I B 0, the mappig ω ξ(ω,i) is a radom variable, whih will be deoted by ξ(i). lso ξ (d) (I) deotes the restritio of ξ(i) om (d).for a Rado measure ω i Γ ad ay I B, defie ξ(ω,i) =ω(i). If P is a probability measure o (M, M), the measure i Γ give by E( ξ(i) )= ξ(ω,i)p (dω),i B, is alled the itesity of P. Similarly if ξ is a measurable futio from some probability spae ito

2 680. Varsei ad. Samimi (M, M), the measure i Γ give by E( ξ(i) ) is alled the itesity of ξ. Fially a system of partitios of Γ (ompare with ull array of partitios i [3] ad see [1]) meas a sequee {Δ } of partitios of Γ havig the followig properties: 1. Eah Δ osists of outable may bouded Borel sets. 2. Δ +1 is a refiemet of Δ for all For ay Γ Γ,Γ B 0, ad ay ε>0, there is some 1, suh that Γ a be overed by a fiite umber of elemets of Δ eah havig diameter smaller tha ε. The topologial struture of Γ esures the existee of suh a sequee of partitios (see [2]). I this artile we prove that if φ ad ψ are two radom measure i Γ defied o (M (d),m (d),p ), with Rado itesity, suh that E( φ(i) M (d) I )=E( ψ(i) M (d) I )P a.s. (1) for all I Δ, where I represets the omplemet of I, the φ = ψ P- a.s. (partially ase was doe i [5] ). 2 Mai Theorem We first represet the followig lemmas. Lemma 1.Ifφ ad ψ are two radom measures i Γ o (M (d), M (d),p) ad satisfyig (1), for ay I Δ ad disjoit I 1,I 2,..., I r Δ ad ay a 1,a 2,..., a r 0: φ(ω, I)P (dω) = ψ(ω,i)p (dω) (2) Where = { ξ(i 1 ) a 1,..., ξ(i r ) a r }. Lemma 2. Two fiite measures μ 1 ad μ 2 o (M (d), M (d) ) with μ 1 ( M (d) )= μ 2 ( M (d) ) are idetial if, ad oly if, for ay disjoit I 1,I 2,..., I r Δ,(ξ(I 1 ) 1,..., ξ(i r ) ) maps μ 1 ad μ 2 to the same measure i [ 0, + ) r. Theorem If φ ad ψ are two radom measures i Γ defied o (M (d), M (d),p), φ, ψ have Rado itesity, suh that E( φ(i) M (d) I )= E( ψ(i) M(d) I )P a.s. for all I Δ, the φ = ψ P- a.s.

3 result o diffuse radom measure 681 Proof. For fixed I Δ, defie a measure o M (d) by μ 1 () = φ(ω, I)P (dω), M(d), ad oe by μ 2 () = ψ(ω,i)p (dω), M(d). Relatio (2), i view of lemma 2, learly implies that μ 1 = μ 2, i.e. φ(ω, I)P (dω) = ψ(ω,i)p (dω) for all M (d). This i tur implies that for P- a.e. ω, Sie that if ω/ N, the φ(ω, I) =ψ(ω,i). Δ is outable, it follows that there is a P-ull evet N suh for all S φ(ω, S) =ψ(ω,s) Δ ad therefore S B. The proof of theorem is thus omplete. 3 Proof of the lemmas Proof of lemma 1: Suppose I, I 0 Δ with I I 0, ad a 0, M (d) I 0, are fixed. Defie, for ν(i),(ν(i) deotes the smallest positive iteger suh that I a be expressed as a uio of elemets of Δ ) X (ω) = χ (ω)φ(ω, J). {ξ(i0 \J) a} Where Δ (I) deotes the family { J Δ : J I }. For eah ω M (d), o aout of its diffuseess, lim max ξ(ω,j) =0.

4 682. Varsei ad. Samimi Therefore lim X (ω) =χ (ω)φ(ω, I). {ξ(i0 ) a} Sie X (ω) φ(ω, I), from the domiated overgee theorem it follows that If we defie, for ν(i), It follows similarly that lim E(χ X )=E(χ Ì φ(i)). {ξ(i0 ) a} Y (ω) = χ {ξ(i0 \J) a} (ω)ψ(ω,j), lim E(χ Y )=E(χ Ì ψ(i)). {ξ(i0 ) a} For eah ν(i), i view of (1) we have It therefore follows that E(χ X )=E(χ Y ). E(χ Ì {ξ(i0 ) a} φ(i)) = E(χ Ì {ξ(i 0 ) a} ψ(i)). (3) Now, if I 1,I 2,..., I r Δ are disjoit, it is ot diffiult to see that we a hoose 0 suh that for eah J Δ (I) 0,orJ Γ\(I 1... Ir ). The = 0,J Ë I s φ(ω, J)P (dω)+ φ(ω, I)P (dω), either J I s for some s, 1 s r 0,J Γ\( Ë I s) φ(ω, J)P (dω)

5 result o diffuse radom measure 683 = 0,J Ë I s ψ(ω,j)p (dω)+ 0,J Γ\( Ë I s) from (3) ad (1), respetively, = ψ(ω,i)p (dω). ψ(ω,j)p (dω) Proof of lemma 2: This lemma is a osequee of the followig lemma whih is osequee of lemma I.2.1 ad Theorem 3.1 of [3]. Lemma 3: Let φ ad ψ be two radom measures i Γ (two measurable futios from some probability spae ito (M, M)). φ ad ψ have the same distributio i M if, ad oly if, for ay disjoit I 1,I 2,..., I r Δ, the radom vetors (φ(i 1 ),..., φ(i r )) ad (ψ(i 1 ),..., ψ(i r )) are idetially distributed. Referees [1] P. Jaob, P.E. Oliveira, O the oditioal itesity of a radom measure 35,1 (1994) [2] O. Kalleberg, O oditioal itesities of poit proesses.z. Wahrsh verw.geb.41 (1987) [3] O. Kalleberg, Radom measure, ademi Press, [4] F. Papagelou., The oditioal itesity of geeral poit proesses ad a appliatio to lie proesses, Z. Wahrsh verw.geb.28(1974) [5] F. Papagelou., Poit proesses o spaees of flats ad other homogeeous Spaes, Math. Pro. Cambridge Pill.So. 80 (1976), [6] S.I. Resik, probability path, Birkhauser Bosto, Reeived: ugust 8, 2006

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