A couple of remarks on the convergence of σ -fields on probability spaces
AA COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDSON PROBABILITY SPACES MATIJA VIDMAR
Abstract.
The following modes of convergence of sub- σ -fields on a given probabilityspace have been studied in the literature: weak convergence, strong convergence, con-vergence with respect to the Hausdorff metric, almost-sure convergence, set-theoreticconvergence, monotone convergence. It is noted that all preserve independence in thelimit, and all are invariant under passage to an equivalent probability measure. Partialresults for the case of operator-norm convergence obtain. Introduction
Fix a (not necessarily complete) probability space p Ω , F , P q and set N : “ P ´ pt uq “t F P F : P p F q “ u . For a sub- σ -field A of F , A P : “ A _ σ p N q “ σ p A Y N q , the P -completion of A . A σ -subfield means a P -complete sub- σ -field of F , i.e. a sub- σ -field of F that is equal to its P -completion. The collection of all σ -subfields is denoted F . For an F { B pr´8 , -measurable map f satisfying ş f ` d P ^ ş f ´ d P ă 8 , P f : “ E P r f s ; if further A P F , then P A f : “ E P r f | A s , the conditional expectation of f w.r.t. A under P .Recall now that for a sequence p B n q n P N Ă F , classical martingale theory gives, for any f P L p P q , the convergence P B n f Ñ P B f in L p P q and P -a.s. as n Ñ 8 , provided one has(MC)
Monotone convergence . B n Ă B n ` for all n P N and B “ _ n P N B n , or B n Ą B n ` for all n P N and B “ X n P N B n [7, Theorem 6.23].Generalizing/complementing this monotone convergence, the following ways of makingprecise the concept of convergence of a sequence of σ -subfields p B n q n P N to a σ -subfield B (under P ), have been proposed and studied in the literature (among others; all theconvergences are as n Ñ 8 ):(WC)
Weak convergence . B n converges weakly to B if P B n A Ñ P B A “ A in P -probability for every A P B [11, 2].(SC) Strong convergence . B n converges strongly to B if P B n A Ñ P B A in P -probability for every A P F [9, 15, 3, 2] [13, Problem IV.3.2] [19, Section 2] [6,Section VIII.2].(HC) Hausdorff convergence . B n converges to B w.r.t. the Hausdorff metric if D p B n , B q Ñ
0, where for A P F and B P F , D p A , B q : “ ρ p A , B q ` ρ p B , A q with ρ p A , B q : “ sup A P A inf B P B P p A (cid:52) B q [4, 17, 12, 14, 20, 10] [6, Section VIII.2]. Mathematics Subject Classification.
Primary: 60A05, 46B28; Secondary: 28A05, 28A20.
Key words and phrases.
Convergence of σ -fields; negligible sets of probability measures; independence. a r X i v : . [ m a t h . P R ] J un COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 2 (STC) Set-theoretic convergence . B n converges to B in the set-theoretic sense iflim inf n Ñ8 B n : “ _ n ě X k ě n B k “ B “ X n ě _ k ě n B k “ : lim sup n Ñ8 B n [5, 1] [16,Problem II.6].(ASC) Almost-sure convergence . B n converges to B in the almost-sure sense if P -a.s. P B n f Ñ P B f for any f P L p P q [1].And, for t p, q u Ă r , , q ď p :(ONC qp ) Operator-norm convergence . B n converges to B in the operator-norm sense if P B n Ñ P B in the operator norm } ¨ } L p Ñ L q when viewed as mappings between the(real) normed spaces p L p p P q , } ¨ } L p p P q q and p L q p P q , } ¨ } L q p P q q [17].Beyond the obvious relevance of these convergence types to the issue of continuity ofconditional expectations w.r.t. the conditioning σ -field, we note applications in statistics[8, 18], studying closeness and convergence of information [20] [6, Section VIII.2], to thetheory of noises [19] (see also the references therein).As for our contribution, we shall demonstrate two desirable properties shared by (es-sentially) all these modes of convergence. First, they all preserve independence in thelimit — and this claim generalizes to conditional independence, save for (WC) — see Sec-tion 3 (in particular Remark 3.2 for the precise meaning of preservation of independencein the limit). Second, excepting (perhaps) only (ONC qp ) when p “ q , all are invariantunder passage to an equivalent probability measure (the latter is trivial for (MC) and(STC), but not obvious for the others) — see Section 4. Given that σ -subfields are ofteninterpreted as bodies of information, and hence convergence of these as a convergenceof information, it is certainly note-worthy that all these types of convergence do in factdepend on the underlying probability measure P only via N “ P ´ pt uq (which, short ofdispensing with the probability measure altogether, is surely the best we can hope for).Likewise, independence is a fundamental probabilistic property – its preservation in thelimit of σ -fields deserves to be made explicit. We will indeed see in relation to this, thatthe simultaneous consideration of the various convergence types enunciated above allowsfor a great economy of argument.We will also show en passant that (ONC qp ) with q ă p is equivalent to (HC), while thecase p “ q “ p “ q “ 8 is vacuous (apart from the trivial case of p B n q n P N ultimatelyconstant), but the case p “ q P p , is not (Section 2).Finally, in terms of what has been noted in the literature in connexion to this thus far: ‚ [19, Corollary 3.6] gives, assuming L p P q is separable, preservation of pairwise inde-pendence in the strong limit. (Were the join operation _ (sequentially) continuous under(SC), then preservation of independence would be an essentially immediate corollary. Butit is not, see e.g. the example of [19, Section 1.2].) ‚ [6, Theorem VIII.2.23] gives invariance of (HC) under passage to a “uniformly abso-lutely continuous” (see [6, Definition VIII.2.22]) probability measure. ‚ [6, Theorem VIII.2.40] gives invariance of (SC) under passage to an equivalent proba-bility measure, assuming L p P q is separable. COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 3 (Note that for r P r , , the separability of L r p P q is equivalent to F “ σ p A q P “ σ p A _ N q for some denumerable A Ă F .) 2. Preliminaries
We gather some relevant results scattered in the literature and make some observations.Let p B n q n P N be a sequence of σ -subfields. (1) Hausdorff metric . Thanks to the insistence on the P -completeness of the σ -subfields, D is a metric on F [4, Theorem 1 & Corollary 1]. By taking _ instead of ` in its definition(see (HC)), one obtains an equivalent metric δ ď ^ D , which is the restriction to F ˆ F of the usual Hausdorff distance on closed subsets of F , associated to the pseudometric F ˆ F Q p
A, B q ÞÑ P p A (cid:52) B q P r , s . (2) Implications and non-implications between the various convergence types .As already observed in the Introduction, (MC) ñ (ASC) (and, of course, (MC) ñ (STC)).(SC) ñ (WC) trivially, but not the other way around — not even when (WC) is tothe largest σ -subfield B for which it holds, cf. (4), second bullet point, below — [9,Example 3.1].(HC) ñ (SC) [4, Theorem 4] and (STC) ñ (SC) [5], though neither conversely [2,Example 4.4]; (ASC) ñ (SC) trivially, again the converse fails [15, Example 3.5].Of (STC), (HC), (ASC) no one implies another: (STC) œ (ASC) [1]; (ASC) œ (STC)[2, Example 4.1]; (STC) œ (HC) [2, Example 4.2]; (HC) œ (STC) [2, Example 4.3]; (HC) œ (ASC) [4, penultimate paragraph]; (MC) (hence (ASC), (STC)) œ (HC) [2, Example4.2].(ONC qp ) ñ (HC). Proof.
For σ -subfields A and B , and for A P A , inf B P B P p A (cid:52) B q “ inf B P B P | A ´ B | ď P | A ´ t P B A ą { u | . Now if q ă 8 , we obtain inf B P B P p A (cid:52) B q ď q P | A ´ P B A | q “ q P | P A A ´ P B A | q ď q } P A ´ P B } q L p Ñ L q , where we have used | A ´ t P B A ą { u | ď q | A ´ P B A | q . When q “ 8 , we have simply inf B P B P p A (cid:52) B q ď P | P A A ´ P B A | ď } P A ´ P B } L Ñ L , since | P A A ´ P B A | ď } P A ´ P B } L Ñ L . (cid:3) Finally, (HC) ñ (ONC qp ), assuming that p ą q . Proof.
Recall the metric δ from (1) and ρ from (HC). We quote the following two resultsfrom the literature:(a) Let a P p , , r P r , , H Ă L r p P q and define δ H,r p a q : “ sup t} f t| f |ą a u } L r p P q : f P H u . Then, for σ -subfields A and B satisfying A Ă B , one has the inequalitysup t} P B f ´ P A f } L r p P q : f P H u ď C r a r δ p A , B qp ´ δ p A , B qqs { r ` δ H,r p a q , where C r “ ¨ { r if r ă C r “ r ě
2. [17, Theorem 4, items (i) & (ii)](b) ρ p A _ B , B q ď ρ p A , B q for σ -subfields A and B . [10, Corollary 4] COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 4 p W C qÒp SC qÕ Ò Ôp HC q p ST C q p
ASC qÙ Ò Õ (ONC qp ) p M C q Figure 1.
Implications between the various types of convergence (with q ă p for the case of (ONC qp )); absence of an (by transitivity implied)arrow means the implication fails in general.Set H : “ t f P L p p P q : } f } L p p P q ď u . For any f P H , by the triangle inequality, } P B n f ´ P B f } L q p P q ď } P B n f ´ P B _ B n f } L q p P q ` } P B n _ B f ´ P B f } L q p P q . Then, by (a), for any a P p , ,sup t} P B n f ´ P B f } L q p P q : f P H uď ¨ { q a r δ p B n , B _ B n qp ´ δ p B n , B _ B n qqs { q ` δ H,q p a q` ¨ { q a r δ p B n _ B , B qp ´ δ p B n _ B , B qqs { q ` δ H,q p a q (where the notation δ H,q p a q is that of (a) above), which by (b) is (note that for σ -subfields A Ą B , δ p A , B q “ δ p B , A q “ ρ p A , B q ) ď ¨ { q aδ p B n , B q { q ` δ H,q p a q . Since δ H,q p a q ď a ´p pq ´ q when p ă 8 and δ H,q p a q ď r , q p a q when p “ 8 , it followsthat lim sup n Ñ8 sup t} P B n f ´ P B f } L q p P q : f P H u ď δ H,q p a q Ñ a Ñ 8 . Hencelim n Ñ8 sup t} P B n f ´ P B f } L q p P q : f P H u “
0, which is the desired operator norm conver-gence. (cid:3) (The last two implications, in the case q “ p “ 8 can be found e.g. in [6, TheoremVIII.2.21].)Up to trivial corollaries, this exhausts the mutual implications and non-implicationsof the convergence types (ONC qp ) for p ą q , (MC), (HC), (STC), (ASC), (SC), (WC)(Figure 1). (3) Uniqueness of limits . Excepting (WC), the limits are unique. Indeed, the results ofe.g. [9] imply uniqueness of the limit in the case of (SC). (4) Weak covergence . ‚ For any p P r , , B n Ñ B weakly iff P B n f Ñ f in L p p P q for every f P L p p P | B q (i.e.for every f P L p p P q that is B measurable). Proof.
By linearity, for sure P B n f Ñ P B f in P -probability for any bounded simple B -measurable f . Now let f P L p p P | B q ; δ ą
0. The simple functions being dense in L p p P q , COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 5 there exists a simple B -measurable f δ for which P | f ´ f δ | p ă δ . Then it follows from thedecomposition P B n f ´ P B f “ P B n p f ´ f δ q ` P B n f δ ´ P B f δ ` P B p f δ ´ f q , from the elementary estimate | x ` y | p ď p ´ p| x | p ` | y | p q for t x, y u Ă R , from condi-tional Jensen’s inequality, finally from the fact that boundedness implies uniform in-tegrability (hence coupled with convergence in P -probability, L p P q convergence), thatlim sup n Ñ8 P | P B n f ´ P B f | p ď Cδ , for some constant C P p , depending only on p .Let δ Ó (cid:3) ‚ Then, according to [2, Lemmas 1.1 and 1.3], B n Ñ B weakly as n Ñ 8 iff B Ă B P , where B P : “ t A P F : lim n Ñ8 inf B P B n P p A (cid:52) B q “ u . The σ -subfield B P coincides with the P - lim inf n Ñ8 B n of [9], see [9, Theorem 3.2]. ‚ The join (sup) operation _ is sequentially continuous under weak convergence [11,Proposition 2.3] (but the meet (inf) ^ is not [11, Proposition 2.1]). It means that forsequences p A n q n P N and p B n q n P N in F , if A n Ñ A and B n Ñ B weakly, then also A n _ B n Ñ A _ B weakly (but in general this fails if X replaces _ ). (5) Strong convergence . With an analogous justification to the one in (4), for any p P r , , B n Ñ B strongly iff P B n f Ñ P B f in L p p P q for every f P L p p P q . Thus strongconvergence is nothing but the strong operator convergence of the conditional expectationoperators in the spaces L p p P q , p P r , (but not in L p P q ; the latter fails even formonotone increasing sequences, see [5, final paragraph]). (6) Operator convergence L p Ñ L p . Convergence in the operator norm } ¨ } L p Ñ L p , p P r , , appears elusive. ‚ For one, (ONC qp ) of not-ultimately-constant sequences of σ -subfields fails always when p “ q “ p “ q “ 8 . Indeed, if A and B are two distinct σ -subfields, then } P A ´ P B } L Ñ L ě } P A ´ P B } L Ñ L ě {
2. To see this, let B P B z A . Set first f “ B ´ P A B , so that 0 ‰ f P L p P q . Then P -a.s., P A f “
0, while f (hence P B f )is ě B and ď z B . It follows that P | P B f ´ P A f | “ P | P B f | “ P rp P B f q B ´p P B f q Ω z B s “ P f B ´ P f Ω z B “ P | f | , viz. } P A ´ P B } L Ñ L ě
1. Set now f “ B , notingthat } f } L p P q “
1. Then P p B (cid:52) t P A B ą { uq ą P p B zt P A B ą { uq ą P pt P A B ą { uz B q ą
0, which coupled with the P -a.s. equality P B f “ B , yields } P A ´ P B } L Ñ L ě { ‚ If for infinitely many n P N , B n Ĺ B or B Ĺ B n — or if B n Ĺ B m for arbitrarily large n P N and m P N — then again P B n does not converge to P B in the } ¨ } L p Ñ L p norm. For if A is a σ -subfield that is strictly contained in the σ -subfield B , one can take B P B z A , set f “ B ´ P A B , which is then not P -a.s. equal to 0, and finds that P -a.s. P B f ´ P A f “ f .In particular, one sees that (ONC qp ) simply precludes (MC) of not-ultimately-constantsequences of σ -subfields altogether. COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 6 ‚ If, for infinitely many n P N , B n contains a non- P -trivial event independent of B or vice versa — or if this obtains with B m in place of B for arbitrarily large m and n , — then P B n does not converge to P B in the } ¨ } L p Ñ L p norm. For if A belongs toa σ -subfield A , is independent of a σ -subfield B , and has P p A q P p , q , then taking f “ P p A q ´ A ´ p ´ P p A qq ´ Ω z A , one has P -a.s. P A f ´ P B f “ f ´ P f “ f , the latternot being P -a.s. equal to 0. ‚ Still, for p P p , , this convergence type is not vacuous with respect to not-ultimately-constant sequences of σ -subfields: Example . Fix p P p , . Let for r P r , s , Ber p r q denote the Bernoulli law (onthe space t , u ) with success parameter r : Ber p r qpt uq “ ´ r and Ber p r qpt uq “ r . LetΩ “ t , u N ; let X i , i P N , be the canonical projections; F the σ -field generated by them; P “ Ber p { q ˆ Ś n P N Ber p { n q . For n P N , set Y n : “ X _ X n and then let B n “ σ p Y n q P ,the P -complete σ -subfield generated by Y n . Note that by the tower property of conditionalexpectations and the independence of the p X m q m P N , for n P N , sup t P | P B n f ´ P B f | p : f P L p p P q , } f } L p p P q ď u “ sup t Q { n | Q { n A f ´ Q { n A f | p : f P L p p Q { n q , } f } L p p Q { n q ď u ,where Q (cid:15) : “ Ber p { q ˆ Ber p (cid:15) q for (cid:15) P r , s , and where, with Z : t , u ˆ t , u Ñ t , u and Z : t , u ˆ t , u Ñ t , u being the projections onto the first and second coordinaterespectively, A : “ σ p Z q and A : “ σ p Z _ Z q . Now fix (cid:15) P r , s . For f P L p Q (cid:15) q , wehave Q (cid:15) A f “ tp , qu f p , q ` tp , q , p , q , p , qu f p , q (cid:15) ` f p , qp ´ (cid:15) q ` f p , q (cid:15) p ` (cid:15) q , whilst Q (cid:15) A f “ tp , q , p , qu r f p , qp ´ (cid:15) q ` f p , q (cid:15) s ` tp , q , p , qu r f p , qp ´ (cid:15) q ` f p , q (cid:15) s . In view of the equality Q (cid:15) | Q (cid:15) A f ´ Q (cid:15) A f | p “ ř ω Pt , uˆt , u Q (cid:15) p| Q (cid:15) A f ´ Q (cid:15) A f | p t ω u q , andusing the elementary estimate p x ` y q r ď r ´ p x r ` y r q for t x, y u Ă r , and r P r , ,it is now straightforward to see that Q (cid:15) r| Q (cid:15) A f ´ Q (cid:15) A f | p s ď C p (cid:15) p p ´ q^ Q (cid:15) r| f | p s for some C p P p , depending only on p . It follows that P B n Ñ P B in the } ¨ } L p Ñ L p operatornorm. ˛ Preservation of independence in the limit
Proposition 3.1.
Let p C n q n P N be a sequence in F with C n Ñ C strongly as n Ñ 8 ; I anarbitrary index set; finally p B in q p n,i qP N ˆ I a collection of σ -subfields of F with B in Ñ B i weakly as n Ñ 8 for each i P I and withthe family p B in q i P I conditionally independent given C n for each n P N .Then the family p B i q i P I is conditionally independent given C .Remark . One says conditional independence is preserved in the limit, under a conver-gence type (iC), when the statement of Proposition 3.1 prevails for any choice of the C n Due to J. Warren, private communication.
COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 7 and the B in s, and with the words “strongly” and “weakly” replaced by “for the convergencetype (iC)” therein. This specializes to (unconditional) independence when C n is P -trivialfor every n P N . By (2) it follows that (unconditional) independence is preserved in thelimit under any of the convergence types (ONC qp )–(MC)–(ASC)–(STC)–(HC)–(SC)–(WC).For conditional independence, we must except (WC), cf. Example 3.3 below. Proof.
We may assume I is finite; then, thanks to (4), last bullet point, via mathematicalinduction and properties of conditional independence, we reduce to the case I “ t , u ;finally, on account of (4), second bullet point, there is no loss of generality in taking B i “ B i P for each i P I . Now take A P B and A P B arbitrary. Then for each i P t , u we find a sequence p A in q n P N with A in P B in for each n P N and with lim n Ñ8 P p A in (cid:52) A i q “ P C n p A X A q Ñ P C p A X A q , P C n p A q Ñ P C p A q and P C n p A q Ñ P C p A q in P -probability. Also, for each n P N , ‚ thanks to independence, P -a.s. P C n p A n X A n q “ P C n p A n q P C n p A n q , ‚ whilst using the elementary equality | C ´ D | “ C (cid:52) D for sets C and D , we findthat P | P C n p A X A q ´ P C n p A n X A n q|“ P | P C n p A X A q ´ P C n p A n X A q ` P C n p A n X A q ´ P C n p A n X A n q|ď P p A (cid:52) A n q ` P p A (cid:52) A n q and likewise P | P C n p A n q P C n p A n q ´ P C n p A q P C n p A q| ď P p A (cid:52) A n q ` P p A (cid:52) A n q . In particular, P C n p A n q P C n p A n q ´ P C n p A q P C n p A q Ñ P C n p A X A q ´ P C n p A n q P C n p A n q Ñ P -probability. Since convergence in probability is preserved un-der addition and multiplication and since the limit in probability is a.s. unique, letting n Ñ 8 , yields that P -a.s. P C p A X A q “ P C p A q P C p A q , as required. (cid:3) Example . We show that conditional independence is generally not preserved under(WC) to the P - lim inf (see (4), second bullet point, for the notation). — Without thelatter insistence, a counterexample is trivial: take any A P F with P p A q P p , q , for n P N let B n “ H : “ σ p A q P be the P -complete σ -field generated by A . Then B n is conditionallyindependent of B n given B n for each n , but the strong (indeed, in every sense) limit H is not conditionally independent of itself given the trivial σ -subfield, to which the B n converge weakly.) — Take Ω “ r , q , F “ B p Ω q the Borel σ -field, P “ Lebesgue measure.For n P N set B n : “ n ´ ´ ď k “ „ k n , k ` n ˙ , and let B n “ σ p B n q P be the P -complete σ -field generated by B n . Finally let B “ tH , Ω u P be the trivial σ -subfield. Then B n converges weakly to B “ B P “ P - lim inf n Ñ8 B n , COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 8 but not strongly [15, Example 3.4]. Finally, denote x “ p? ´ q{
8; notice that 1 { ă ´ x ` { ă ´ x ă ´ x ` ă
1; and consider the events A : “ „ , ˙ Y r ´ x, q and B : “ „ , ˙ Y „ ´ x ` , ´ x ` ˙ so that A X B “ „ , ˙ Y „ ´ x, ´ x ` ˙ . A simple calculation yields that P -a.s. for each n P N ě , P p A | B n q “ P p B | B n q “ { { B n ` { ` x { Ω z B n whilst P p A X B | B n q “ { { B n ` { ` { { Ω z B n . This renders the P -a.s. equality P B n p A q P B n p B q “ P B n p A X B q . Thus A and B , equiva-lently the respective P -complete σ -fields generated by them, are conditionally independentgiven B n for each n P N ě . But A and B are not conditionally independent given the weaklimit B “ B P of the B n , for they are not independent: p { ` x q ‰ { ` {
16, as isreadily verified. ˛ Invariance under passage to equivalent probability measure
In what follows, for a convergence mode (iC), by saying that it is invariant under passageto an equivalent probability measure, we mean that, whenever p B n q n P N Ă F , B n Ñ B in the sense of (iC) under P ðñ B n Ñ B in the sense of (iC) under Q ,provided Q „ P . (Note that, given p Ω , F q , F depends on P only through N “ P ´ pt uq .)Recall that for finite measures µ and ν on p Ω , F q , µ ! ν is equivalent to @ (cid:15) P p ,
8q D δ P p ,
8q @ A P F : ν p A q ă δ ñ µ p A q ă (cid:15). Proposition 4.1. (HC) is invariant under passage to an equivalent probability measure.Indeed the distance D , up to equivalence, depends on P only up to equivalence.Remark . By (2), (ONC qp ), q ă p , is equivalent to (HC), so that (ONC qp ), q ă p , toois invariant under passage to an equivalent probability measure. The case p “ q remainsopen. Proof.
Let Q „ P , denote by D P and D Q the metrics associated to P and Q , respectively.Let (cid:15) P p , , A P F . Since Q ! P , there is a δ P p , such that for all A P F , P p A q ă δ ñ Q p A q ă (cid:15) . Then for all B P F , D P p A , B q ă δ implies D Q p A , B q ď (cid:15) . (cid:3) Proposition 4.3. (WC) is invariant under passage to an equivalent probability measure.Proof.
By (4), second bullet point, it is enough to verify that B P “ B Q whenever P „ Q .But assuming Q ! P , thanks to the equivalent condition for absolute continuity notedabove, if for some A P F there exist A n P B n for n P N with lim n Ñ8 P p A n (cid:52) A q “
0, thenalso lim n Ñ8 Q p A n (cid:52) A q “ (cid:3) COUPLE OF REMARKS ON THE CONVERGENCE OF σ -FIELDS ON PROBABILITY SPACES 9 Recall now the statement of abstract Bayes’ theorem. Letting Q be another probabilitymeasure on F , equivalent to P :For any F / B pr´8 , -measurable X and any sub- σ -field G of F , Q G p X q P G p d Q d P q “ P G p d Q d P X q a.s., in the sense that the left hand-side is well-defined iff the right hand-side is so, whence they are equal. Proposition 4.4. (SC) is invariant under passage to an equivalent probability measure.Proof.
Assume B n Ñ B strongly under P . Let Q „ P . Note d Q { d P and then all the P B n p d Q d P q , n P N , may be chosen from their equivalence classes to be strictly positiveeverywhere. Let A P F . Then by Bayes’ rule a.s. Q B n A “ P B n ˆ d Q d P A ˙ M P B n ˆ d Q d P ˙ . By (5) the numerator and denominator both converge as n Ñ 8 in L p P q , hence in P -(equivalently, Q -) probability, to the respective expressions in which B replaces B n . Con-vergence in probability is preserved under taking quotients (assuming the denominatorsare non-zero; e.g. from the characterization through the a.s. convergence of subsequences)and the claim follows by another application of Bayes’ rule. (cid:3) Remark . According to [2, Proposition 3.3, Lemma 1.3] (SC) is equivalent to the con-junction of (WC) and( K C) Orthogonal convergence . B n converges to B orthogonally if A n ´ P B A n Ñ p P q , whenever A n P B n for each n P N .It remains open whether ( K C) too is invariant under passage to an equivalent probabilitymeasure.
Proposition 4.6. (ASC) is invariant under passage to an equivalent probability measure.Proof.
Assume B n Ñ B in the almost-sure sense under P . Let Q „ P . Then by Bayes’rule, for any f P L p Q q , a.s. Q B n f “ P B n ˆ d Q d P f ˙ M P B n ˆ d Q d P ˙ . Since d Q d P f and d Q d P both belong to L p P q , by the very definition of (ASC), the numeratorand denominator both converge P - (equivalently Q -) a.s. as n Ñ 8 to the respectiveexpressions in which B replaces B n . Another application of Bayes’ rule concludes theargument. (cid:3) Question . Given this invariance of the various convergence modes, can somethingakin to the characterization of convergence in probability through the a.s. convergence ofsubsequences, be offered? I.e. can the convergence modes be characterized in such a wayas to make manifest the invariance under passage to an equivalent probability measure?
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