Constituting Atoms of a $sigma$ Algebra via Its Generator

Constituting A toms of a σ Algebra via Its Generator Jinshan Zhang ∗ Department of Mathematical Sciences, Tsinghua Universit y , Beijing 1000 84, China Abstract In this pap er, a v ery weak sufficien t cond ition for determinin g atoms by the gen- erator is presented. The condition, though not b eing a necessary one, is shown to b e almost the w eak est one in the sense that it can hard ly b e impro ved. Mathematics Sub ject Classification (2000). Primary 03E20 ; S econdary 31C99. Keyw ords: at om; σ algebra; generator; monotone class; κ class. 1 In tro duction The atom is an adv anced concept in m o dern probabilit y theory . An in tensive application of the prop er ties of atoms pla ys an imp ortant role in one of the p ro ofs of the famous V atali-Hahn-Saks Theorem [7]. This concept is essen tial in the deve lop- men t of conditional probabilit y during the recent decades[1, 3, 5, 6]. How ev er wh en atom is used, the relations b et ween σ algebra and its atoms is mostly assumed to b e kno wn . Namely most of the authors consider suc h relations as conditions in their researc h. This p ap er fo cu s es on ho w to determine s u c h relations, whic h means to find the structures and the constructions of atoms of a σ algebra. Th e most relev ant result to this pap er should b e Blac kw ell theorem [4], which is quite u seful an d has reduced inclusion b et ween Blac kwell σ algebras to comparing their atoms. He nce, determining the atoms of a σ alge bra b ecomes so signifi can t wh en app lyin g th at p o w erf ul theorem to solve p roblems. The σ algebra itself, ho wev er, is u sually to o large or can not b e efficien tly ob- tained b y giv en its generator only . T o solv e this matter, red u cing co nstituting atoms ∗ Electronic address: zj s02@mails.tsinghua. edu.cn 1 from the σ alg ebra to constituting them from the generator would b e a feasible and efficien t app r oac h. Therefore, at least some sufficien t conditions for the generator on its structure, whic h can b e used to constitute the atoms of the σ algebra generated, should b e provided. The condition for the generator pr op osed in this pap er only requires κ ( C ) = σ ( C ). This condition, though not b eing a necessary one, is sh o wn to b e almost the weak est one in the sense that it can hardly b e im p ro ved. The follo wing is a br ief road-map of the pap er. In section 2, some preliminary kno wledge is introd uced, whic h ma y b e used in this pap er. W e go on in section 3 to pro ve the main theorem of the pap er. S ome useful corolla r ies follo wing the theorem construct the sub ject of section 4. In s ection 5, the condition in the main theorem is d iscussed through sp ecific examples and th eoretical analysis. Concluding remarks are prop osed. 2 Preliminaries In this section we review some fun damen tal concepts and relat ed resu lts that w ould b e utilized in this p ap er. They are mostly imp ortan t definitions and theorems in probabilit y theory . In the follo w ing, we alw ays let Ω b e a set, and C b e a collection (set) of s u bsets of the set Ω. Definition 2.1 C is called a monotone class, if A n ∈ C , n ≥ 1 , A n ↑ A or A n ↓ A ⇒ A ∈ C . F urther, C is called a λ class, if C is a monotone class, Ω ∈ C , and ∀ A , B ∈ C , B ⊂ A ⇒ A ∩ B c ∈ C . Definition 2.2 Let (Ω , F ) b e a measur able space, where F is a σ algebra on Ω. F or an y ω ∈ Ω define F ω = { B ∈ F | ω ∈ B } . T hen A ( ω ) = T B ∈F ω B is called an atom of F contai ning ω . Let C ∩ f = { A | A = n T i =1 A i , A i ∈ C , i = 1 , · · · , n, n ≥ 1 } b e the set closed under finite intersectio n . S imilarly , C ∪ f , C P f , C δ , C σ , C P σ denote the sets closed u nder finite u nion, fin ite disjoint union, countable intersect ion, coun table union, coun table disjoin t u nion r esp ectiv ely . C σδ = ( C σ ) δ . σ ( C ), λ ( C ), m ( C ) denotes the minimum σ 2 algebra, λ class, and monotone class con taining C resp ectiv ely and C is the generator of them. 3 Main theorem Let C b e a collection of sets on Ω and F = σ ( C ). F or an y ω ∈ Ω, define C ω = { B ∈ C | ω ∈ B } and A ( ω ) the atom co n taining ω of F . Question is: und er wh at condition on C , there w ould b e A ( ω ) = \ B ∈C ω B . Y an [7] sho w s that this is true, if C is an alge bra. A m uch w eak er condition on C is prop osed in this section, wh ic h is the main result of this p ap er. In order to show the main result of this pap er, w e in tro duce the follo wing concept, whic h is created in this pap er to s ho w how go o d our condition is. Definition 3.1 C is called a κ class, if it is closed under coun table intersecti on and coun table union. Denote b y κ ( C ) the minimum κ class con taining C , and C is called generato r of κ ( C ). T o complete the p ro of of the m ain r esult, w e n eed the follo wing lemma. Lemma 3.1 L et C b e a collec tion of sets on Ω, F = σ ( C ). ∀ ω ∈ Ω, define C ω = { B ∈ C | ω ∈ B } and A C ( ω ) = T B ∈C ω B . Let G = { B ∈ F | ω / ∈ B , or ω ∈ B and A C ( ω ) ⊂ B } = { B ∈ F | ω / ∈ B } S { B ∈ F | ω ∈ B , A C ( ω ) ⊂ B } . Then G satisfies the follo win g three prop erties: 1). C ⊂ G , C σ ⊂ G . 2). G is closed un der the op eration of counta b le union and coun table intersect ion. 3). G is a κ class. In particular, it is a monotone cla ss . Pro of: Let C σω = { B ∈ C σ | ω ∈ B } and A C σ ( ω ) = T B ∈C σ ω B . Claim ∀ ω ∈ Ω, A C ( ω ) = A C σ ( ω ). First, C ω ⊂ C σω , then A C σ ( ω ) = T B ∈C σ ω B ⊂ T B ∈C ω B = A C ( ω ). Con- sider the definition of C σ and C σω , ∀ B ∈ C σω , ∃ { A n } ∞ n =1 ⊂ C su c h that B = ∞ S n =1 A n . Hence, there exits N such that ω ∈ A N , then there are A C ( ω ) ⊂ A N ⊂ B and A C ( ω ) ⊂ T B ∈C σ ω B = A C σ ( ω ). Hence A C ( ω ) = A C σ ( ω ). Now, let’s prov e the lemma. 3 F or the prop ert y 1. ∀ B ∈ C , if ω / ∈ B then B ∈ G ; Otherwise, if ω ∈ B , since A C ( ω ) = T B ∈C ω B , w e ha ve A C ( ω ) ⊂ B , then B ∈ G . Hence C ⊂ G . F ro w the claim, w e k n o w G = { B ∈ F | ω / ∈ B , or ω ∈ B and A C σ ( ω ) ⊂ B } = { B ∈ F | ω / ∈ B } S { B ∈ F | ω ∈ B , A C σ ( ω ) ⊂ B } . Thus, similarly , C σ ⊂ G . F or the pr op ert y 2. Su pp ose { A n } ∞ n =1 ⊂ G . (i). If ∀ n , ω / ∈ A n , then ω / ∈ ∞ S n =1 A n . Hence ∞ S n =1 A n ∈ G . (ii). If ∃ n suc h that ω ∈ A n then A C ( ω ) ⊂ A n ⊂ ∞ S n =1 A n . Obviously , ω ∈ ∞ S n =1 A n . Hence ∞ S n =1 A n ∈ G Considering (i) and (ii), G is closed und er coun table union. (iii). If ∃ n suc h that ω / ∈ A n , then ω / ∈ ∞ T n =1 A n . Th u s ∞ T n =1 A n ∈ G . (iv). If ∀ n , ω ∈ A n , then ω ∈ ∞ T n =1 A n . Since A C ( ω ) ⊂ A n ( ∀ n ), A C ( ω ) ⊂ ∞ T n =1 A n . Hence ∞ T n =1 A n ∈ G . Considering (iii) and (iv), G is closed under coun table in tersection. F or prop ert y 3. F rom prop erty 2, we kno w G is a κ cla ss. In particular, if A n ↑ A then A = ∞ S n =1 A n , and if A n ↓ A , then A = ∞ T n =1 A n . Hence G is a monotone class. ✷ Using Lemma 3.1, n o w we prov e the main r esult of th is pap er. Theorem 3.1 Let C b e a collection of sets on Ω , F = σ ( C ) and A F ( ω ) the atom of F con taining ω . ∀ ω ∈ Ω, defin e C ω = { B ∈ C | ω ∈ B } and A C ( ω ) = T B ∈C ω B . If the generator C satisfies the pr op ert y that ∀ A ∈ C ⇒ A c ∈ κ ( C ), then A F ( ω ) = A C ( ω ) . Pro of: ∀ ω , let G 1 = { B ∈ F | ω / ∈ B , or ω ∈ B and A C ( ω ) ⊂ B } and G 2 = { A ∈ G 1 | A c ∈ G 1 } . Then G 2 satisfies the f ollo wing pr op erties. (a). ∀ A ∈ C , A c ∈ κ ( C ), by the prop ert y 1 and 3 of G 1 in L emma 3.1, w e kno w κ ( C ) ⊂ G 1 , then A c ∈ G 1 . Hence C ⊂ G 2 . (b). Since G 1 is a monotone cla s s, it’s easy to chec k G 2 is a monotone class. (c). No w let’s c hec k G 2 is an algebra. (i). ∀ A ∈ G 2 , then A ∈ G 1 , A c ∈ G 1 , ( A c ) c ∈ G 1 , h ence A c ∈ G 2 . (ii). ∀ A , B ∈ G 2 , then A , A c ∈ G 1 and B , B c ∈ G 1 . Consider the p rop erty 2 of G 1 , 4 w e kn o w A ∩ B ∈ G 1 , A c ∪ B c ∈ G 1 , then ( A ∩ B ) c ∈ G 1 . Hence A ∩ B ∈ G 2 . Considering (i) and (ii), we sh ow G 2 is an algebra. Now from (a), (b) and (c) , G 2 is a monotone class and algebra con taining C . By Monotone Class Theorem, F = σ ( C ) ⊂ G 2 . Then G 2 = F . G 2 ⊂ G 1 ⊂ F , then G 1 = F . Noting that G 1 / F ω = { B ∈ F | ω / ∈ B } (recall F ω = { B ∈ F | ω ∈ B } ), then ∀ B ∈ F ω A C ( ω ) ⊂ B . Hence A C ( ω ) ⊂ A F ( ω ). since C ω ⊂ F ω , A F ( ω ) ⊂ A C ( ω ). Th u s the result of this theorem follo ws. ✷ 4 Corollaries In this section useful corollaries follo w ing the main theorem is present ed. Corollary 4.1 If C is a semi-algebra on Ω, and F = σ ( C ). Th en ∀ ω ∈ Ω, A C ( ω ) = A F ( ω ) . Pro of: F or an y A ∈ C , one has A c = Ω / A ∈ C P f ⊂ C σ ⊂ κ ( C σ ) = κ ( C ) . Hence the result f ollo ws. ✷ Corollary 4.2 If C is a semi-ring, Ω ∈ C σ , F = σ ( C ). T h en ∀ ω ∈ Ω , A C ( ω ) = A F ( ω ) . Pro of: Th ere exists a sequence A n ∈ C s uc h that Ω = ∞ S n =1 A n . Then A c = ∞ [ n =1 ( A n / A ) , b y noting A n / A ∈ C P f ⊂ C σ . Hence A c ∈ C σ ⊂ κ ( C ). ✷ Corollary 4.3 If κ ( C ) = σ ( C ), then ∀ ω ∈ Ω, A C ( ω ) = A F ( ω ). Pro of: W e sho w the equiv alence b etw een κ ( C ) = σ ( C ) and ∀ A ∈ C ⇒ A c ∈ κ ( C ). If κ ( C ) = σ ( C ), obvio u sly , th ere are ∀ A ∈ C ⇒ A c ∈ κ ( C ). F or the in verse direction, the collection of set G = { B ∈ κ ( C ) | B c ∈ κ ( C ) } , w hic h is closed u nder counta b le 5 in tersection, coun table union and complement, con tains C . Hence, G is a σ algebra and G = κ ( C ) = σ ( C ). ✷ . In the follo w ing corollaries w e sup p ose that F is separable( F can b e generated b y a coun table subset), so they can b e d ir ectly applied to the comparison among atoms in Blac kwell sp ace[4]. Corollary 4.4 Supp ose C is a coun table semi-ring and F = σ ( C )(Obviously , F is separable). Then ∀ ω ∈ Ω, A C ( ω ) = A F ( ω ) if and only if Ω ∈ C σ . Pro of: F rom C orollary 4.2, w e know we only ha ve to chec k if ∀ ω ∈ Ω A C ( ω ) = A F ( ω ) ⇒ Ω ∈ C σ . ∀ ω ∈ Ω A C ( ω ) = A F ( ω ), then ∃ B ∈ C suc h that ω ∈ B . Hence, Ω = S B ∈C B . Note C is coun table, th en Ω = S B ∈C B ∈ C σ . ✷ Corollary 4.5 Let C b e a collecti on of sets on Ω, F = σ ( C ). If F has coun t- able atoms and C is coun table. Then ∀ ω ∈ Ω, A F ( ω ) = A C ( ω ) if and only if F = κ ( C ). Pro of: If ∀ ω ∈ Ω, A F ( ω ) = A C ( ω ). ∀ A ∈ C , A c = S ω ∈ A c A F ( ω ) = S ω ∈ A c A C ( ω ). Since C is coun table, A C ( ω ) = T B ∈C ω B ∈ κ ( C ). Since the atoms of F is count able, S ω ∈ A c A C ( ω ) = S ω ∈ A c T B ∈C ω B ∈ κ ( C ), ind icating A c ∈ κ ( C ). F r om the pro of of Corol- lary 4.3, A c ∈ κ ( C ) ( ∀ A ∈ C )implies F = κ ( C ). Th e in verse that F = κ ( C ) implies ∀ ω ∈ Ω A F ( ω ) = A C ( ω ) is tr ivial if we n ote that F = κ ( C ) implies ∀ A ∈ C , A c ∈ κ ( C ). ✷ In Corollary 4.1 and 4.2, we do not really use the p rop erty of s emi-rin g or semi- algebra, whic h is closed und er finite in tersection. Besides the condition A ∩ B c ∈ C P f can b e replaced by A ∩ B c ∈ C σδ . 5 Discussion and conclusion First consider the follo wing t w o examples. Example 5.1 Let Ω = R , C = { x | x ∈ R } and F = σ ( C ). Obvio u sly , ∀ x ∈ R A C ( x ) = x = A F ( x ), and F is Hausdoff(the atoms of F are the p oin ts of Ω). It’s 6 easy to chec k κ ( C ) ⊂ { A ⊂ R | A is coun table } . How ev er, R/ { 0 } ∈ F is not in κ ( C ). This sho w s ou r condition is not a necessary one. Example 5.2 L et Ω = [0 , 1], C = { [ a, b ) ⊂ [0 , 1) | a < b } ∪ {∅} and F = σ ( C ). C is a semi-ring on [0,1]. A C (1) = ∅ , while A F (1) = { 1 } . This sho w s the condition that the generator is a semi-ring is not sufficien t for A C ( ω ) = A F ( ω )( ∀ ω ∈ Ω). The examples sho w that our condition ma y not b e the b est one b ut almost nec- essary . Comparing our condition with semi-ring (see Corollary 4.2), w e only add Ω ∈ C σ to obtain the desired result, and the co ndition of Corollary 4.2 is stronger than that in our main theorem. Th erefore our condition has already b een a very w eak one. On the other hand since m ( C ) ⊂ m ( C σ ) ⊂ κ ( C σ ) = κ ( C ) ⊂ σ ( C ) and m ( C σ ) ⊂ λ ( C σ ) ⊂ σ ( C ). T he tr ivial case A c ∈ σ ( C ) cont ributes nothing if letting it replace A c ∈ κ ( C ) since it is im p ossible to conclude A C ( ω ) = A F ( ω )( ∀ ω ∈ Ω) without any restriction on C (Example 5.1 can b e view ed as a sp ecial coun terexam- ple). F rom relations among m ( C ), m ( C σ ), κ ( C σ ), κ ( C ), λ ( C σ ), σ ( C ), w e kno w κ ( C ) is already a v ery large set and n early as large as λ ( C σ ). Finally , rewiewing the pro of of the theorem, one can find the k ey of the pr o of lies in the pr op ert y of G in L emm a 3.1. Generally , G is at most a κ class and could not b e a λ class. Hence th e imp r o veme n t of our condition from the theoretic p ersp ectiv e is almost imp ossible. Ac kno w ledgemen ts. The author is v ery grateful to P rofessor Yve s Le Jan for helpful discussions and bringing the reference [4 ] to his attent ion. References [1] D. Blac kwell and L. E. Du bins. 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