A ZERO-ONE LAW FOR RIESZ SPACE AND FUZZY PROCESSES ∗ Wen-Chi Kuo1

Coenraad C.A. Labuschagne1 1. School of Mathematics

Bruce A. Watson1

University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa Email: [email protected], [email protected], [email protected]

Abstract

an alternative approach. A fundamental concept which has, to our knowledge, not appeared previously in the Riesz space context, is independence. Here we introduce the notions of independence of both families from the Riesz space and for band projections with repect to a given conditional expectation operator. In should be noted that in the classical probability setting, independence is with respect to one of the simplest conditional expectation operators, i.e. the expectation operator. The closely linked concept of the tail of a sequence is defined next. From an independent sequence with respect to a given conditional expectation, we are able to show the tail to be contained in the range space of the contitional expectation operator. Hence we obtain Riesz space and fuzzy analogues of the Borel-Cantelli Lemma and of the Kolmogorov Zero-One Law. We conclude the paper by indicating the relevance of our results to classical and fuzzy probability. Section 2 provides the necessary background on conditional expectation operators acting on Riesz spaces. The concept of independence relative to a conditional expectation operator in a Riesz space is explored in Section 3. Riesz space versions of the Borel-Cantelli Lemma and the Kolmogorov zero-one law are proved in Section 4. In Section 5 the application of our results to both classical (measure theoretic) and fuzzy probability theory is discussed.

In Discrete-time stochastic processes on Riesz spaces, Indag. Mathem., N.S., 15(3), 435-451, we introduced the concepts of conditional expectations, martingales and stopping times on Riesz spaces. In the present work we introduce the concept of independence on a Riesz space with a conditional expectation operator. This concept coincides with the probabilistic definition in the case of the conditional expectation operator being an expectation operator. For a Riesz space with conditional expectation operator we prove analogues of the BorelCantelli lemma and the Kolmogorov zero-one law. We conclude by showing the relevance of our results to both classical and fuzzy probability theory. It should be noted that in the Riesz space setting, the space of random variables does not necessarily possess an expectation operator, but does have a conditional expectation operator. Keywords: IFSA2005, Riesz space, Fuzzy, Zeroone, Conditional expectation.

1 Introduction

Conditional expectations have been studied in an operator theoretic setting, by de Pagter, Dodds, Grobler, Huijsmans and Rao, see [2, 3, 8], as positive operators acting on L p -spaces and Banach function spaces. The concepts of conditional expectations, martingales 2 Preliminaries and stopping times on Dedekind complete Riesz space with weak order units were introduced in [4], see [9] for The reader is assumed familiar with the notation and terminology of Riesz spaces, for details see [11]. ∗ WK and BAW funded in part by the Centre for Applicable Analysis and Number Theory, and WK funded in part by the VCO of the University of the Witwatersrand.

Definition 2.1 Let E be a Riesz space with weak order unit. A positive order continuous projection T : E → E, 1

with range R (T ) a Dedekind complete Riesz subspace Proof: To prove the result we need only show that f ∈ of E, is called a conditional expectation if T (e) is a BT f . Let weak order unit of E for each weak order unit e of E. g = (I − PT f )e ∈ R (T )+ . It can be shown that if T is a conditional expectation Then as (T f ) ∧ g = 0 we have on E then there exists a weak order unit e with e = Te, 0 ≤ T ( f ∧ g) ≤ T f ∧ T g = (T f ) ∧ g = 0. see [4] for details. Note, if E = L1 (Ω, F , P) is a probability space and Σ From the strict positivity of T , f ∧ g = 0. Hence is a sub-σ-algebra of F , then E is a Dedekind complete Riesz space with weak order unit e = 1 and f ∧ (I − PT f )e = 0 T f = E[ f |Σ]

so that

is a Riesz space conditional expectation operator on E Thus with Te = e, see [4, 5]. Let E be a Dedekind complete Riesz space with weak order unit, say e. If f is in the positive cone,

(I − PT f ) f = 0. f = PT f f ∈ BT f .

E + := { f ∈ E | f ≥ 0},

3 Independence

of E then the band generated by f is

In probability theory the concept of independence relies on both the presence of a probability measure and the multiplicative properties of R+ . In the Riesz space setting, the role of the probability measure or expectation operator is replaced by a conditional expectation operator while the role of multiplication is mirrored at operator level by composition.

B f = {g ∈ E | |g| ∧ n f ↑n |g|}. Let Pf g = sup g ∧ n f ,

for all g ∈ E + ,

n

then Pf can be uniquely extended to E by setting P f g = P f g+ − P f g− .

Definition 3.1 Let E be a Dedekind complete Riesz space with conditional expectation T and weak order unit e = Te. Let P and Q be band projections on E, we say that P and Q are independent with respect to T if

This map Pf is then a band projection onto B f . A consequence of this is that in a Dedekind complete Riesz space with weak order unit e, if P is a band projection onto a band B, then B is the principal band generated by Pe. We recall the following result from [6], needed later.

T PT Q = T PQ = T QT P.

In the case of E = L1 (Ω, A , µ) where µ is a probabilTheorem 2.2 Let T be a conditional expectation on a ity measure, e = 1 and T an expectation operator Dedekind complete Riesz space, E, with weak order unit Z and let P be the band projection of E onto the band, B, T f = f dµ = E[ f ]1, in E generated by 0 ≤ g ∈ R (T ). Then T P = PT . An interesting consequence of Theorem 2.2 is an av- we have that the band projections on E are maps of the eraging property of T which is manifest even in the ab- form P f = f χA and Q f = f χB where A, B ∈ A . Here sence of a multiplicative structure, as seen in the folT PT Qe = E[χA E[χB ]] = E[χA µ(B)] = µ(B)E[χA] = µ(B)µ(A) lowing corollary. Corollary 2.3 Let E be a Dedekind complete Riesz and similarly space with strictly positive conditional expectation opT QT Pe = µ(A)µ(B). erator T and weak order unit e = Te. Denote by B f the band in E generated by f ≥ 0 and by Pf the band pro- Also jection of E onto B f . Then for each f ∈ E + we have Pf ≤ PT f . T PQe = E[χA χB ] = E[χA∩B ] = µ(A ∩ B). 2

Definition 3.3 Let E be a Dedekind complete Riesz space with conditional expectation T and weak order unit e = Te. We say that the sequence ( fn ) in E is independent with respect to T if the family

Thus, in this case, the Riesz space independence of P and Q corresponds to the classical independence of A and B. A more interesting example is that of, say, E = L1 ((0, 1), A , µ) where µ is Lebesgue measure, A the Lebesgue measurable sets and T f = 2χ(0,1/2]

Z 1/2 0

f dµ + 2χ(1/2,1)

Z 1

{h{ fn , e}i |n ∈ N} of Dedekind complete Riesz spaces is independent.

f dµ.

1/2

A linear map R : E → E is said to be strictly positive if R f > 0 for all 0 < f ∈ E. The following lemma forms a crucial step in the proof of the zero-one laws.

Here T is a conditional expectation on E and e = 1 is a weak order unit which is invariant under T . In this case the Riesz space independence of P and Q (as above) gives that A and B are independent with respect to both of the conditional probability measures µ(0,1/2] and µ(1/2,1]. In a manner similar to that of measure theoretic probability, we can define independence, with respect to the conditional expectation T , of a family of Dedekind complete Riesz subspaces of the Riesz space E.

Lemma 3.4 Let E be a Dedekind complete Riesz space with strictly positive conditional expectation T and weak order unit e = Te. Let P be a band projection on E which is self-independent with respect to T , then T P = PT and T Pe = Pe. Proof: As P is self-independent with respect to T , we have that T PT P = T P. Thus

Definition 3.2 Let E be a Dedekind complete Riesz 0 = T (P − PT P) = T (I − P)T P, space with conditional expectation T and weak order unit e = Te. Let Eλ , λ ∈ Λ, be a family of Dedekind since T 2 = T . Because I − P ≥ 0 and T strictly positive, complete Riesz subspaces of E having e ∈ Eλ for all it now follows that λ ∈ Λ. We say that the family is independent with respect to T if, for each pair of disjoint sets Λ1 , Λ2 ⊂ Λ, (I − P)T P = 0 we have that P1 and P2 are independent with respect to and hence T . Here P1 and P2 are band projections with + * PT P = T P. (3.1) [ Pj e ∈ Eλ , Direct computation gives λ∈Λ j T PT (I − P) = T PT − T (PT P) = T PT − T P.

where hSi denotes the smallest Dedekind complete Riesz subspace of E containing the set S.

Since Te = e, (3.2) when applied to e gives

For E = L1 (Ω, A , µ), where µ is a probability measure, e = 1 and T the expectation operator Tf =

Z

(3.2)

T PT (I − P)e = 0, from which the strict positivity of T allows us to deduce that PT (I − P)e = 0.

f dµ = E[ f ]1,

independence of the family,

Combining this result with (3.1) and Te = e yields

Eλ = L1 (Ω, A λ , µ), λ ∈ Λ,

Pe = PTe = PT Pe = T Pe.

of Dedekind complete Riesz sub-spaces Eλ of E, where A λ is a sub-σ-algebra of A , is none other than the definition of independence of the family A λ , λ ∈ Λ, as subσ-algebras of A . Definition 3.2 leads naturally to the definition of independence for sequences in E, given below.

Let B be the band associated with P. Then B is the band generated by Pe, as remarked at the end of Section 2. Since T Pe = Pe, B is a band generated by an element of R (T ), and thus from [5, Lemma 3.1] it follows that T P = PT . 3

4 Zero-One Law

thereby proving the theorem.

Note that if P is a band projection, then 0 ≤ P ≤ I. In this section we formulate and prove, in a Riesz space Let (Pj ) be a sequence of band projections in E. Hence setting, variants of the Borel-Cantelli Lemma and the Kolmogorov Zero-One Law. The presence of neither a Q := lim sup Pj probability measure nor an expectation operator is asj→∞ sumed, but this role is taken by the conditional expectation operator. The use made of the multiplicative struc- exists and is again a band projection. In fact, ture of R+ in the classical setting will now be filled by lim sup Pj e j→∞ the composition of band projections, as seen below. Lemma 4.1 Let E be a Dedekind complete Riesz space is a weak order unit for the band associated with Q. Takwith conditional expectation operator T and weak or- ing f j = Pj e in the above lemma, we have as a straight forward corollary, the following Riesz space analogue der unit e = Te. If of the classical Borel-Cantelli Theorem. ∞

∑ T f j ∈ E,

Corollary 4.2 Borel-Cantelli Let E be a Dedekind complete Riesz space with strictly positive conditional expectation operator T and weak order unit e = Te. If

j=1

where ( f j ) is an order bounded sequence in E + , then ! T

∞

∑ T Pj e ∈ E,

= 0.

lim sup f j j→∞

j=1

where (Pj ) is a sequence of band projections in E, then

If, in addition, T is strictly positive then

lim sup Pj = 0.

lim sup f j = 0.

j→∞

j→∞

In order to consider the Kolmogorov Zero-One Law, Proof: The boundedness of the sequence ( f j ) ensures we need a definition for the tail Riesz subspace of seexistence of all suprema and limits involved in the quences in a Riesz space. This definition of the tail proof. As each f j ≥ 0, it follows that must generalize the classical σ-algebra definition for a sequence of random variables. sup f j ≤ ∑ f j , for all k ∈ N. j≥k j≥k Definition 4.3 Let E be a Dedekind complete Riesz space with weak order unit e. Let ( fn ) be a sequence From the order convergence of ∑ T f j we get in E. We define the tail of ( fn ) with respect to the weak order unit e to be the Dedekind complete Riesz subspace ∑ T f j → 0, in order as k → ∞. of E given by j≥k τ[( fn ), e] :=

Combining these results gives ! 0≤T

sup f j j≥k

≤

h{e, fn , fn+1 , . . .}i .

n∈N

∑ T f j → 0,

We say that a band projection P is from the tail of ( f j ) if Pe ∈ τ[( f j ), e].

in order as k → ∞.

j≥k

Thus

\

!

Theorem 4.4 The Kolmogorov Zero-One Law Let E be a Dedekind complete Riesz space with strictly k→∞ j≥k positive conditional expectation operator T and weak and the order continuity of T allows us to conclude that order unit e = Te. Let ( fn ) be a sequence in E which is ! ! independent with respect to T . If P is a band projection   T lim sup f j = T lim sup f j = lim T sup f j = 0, from the tail of ( fn ) with respect to e, then T Pe = Pe k→∞ k→∞ j≥k and PT = T P. j≥k k→∞ lim T

sup f j

= 0,

4

Proof: From Lemma 3.4, we need only prove that P is self-independent with respect to T . For each n ∈ N let Sn

=

h{e, f1 , . . . , fn }i ,

S∞

=

h{e, f1 , f2 , . . .}i .

Corollary 4.5 Let E be a Dedekind complete Riesz space with strictly positive conditional expectation operator T and weak order unit e = Te. Let ( fn ) be a sequence in E independent with respect to T , then τ[( fn ), e] ⊂ R (T ).

As P is a band projection from the tail of ( fn ) we have that Pe ∈ S∞ . But

∞ [

Proof: From Theorem 4.4, if P is a band projection with

S j is order dense in S∞ and conse-

Pe ∈ τ[( fn ), e]

j=1

quently there exists a net (hα ) ⊂

∞ [

then Pe = T Pe ∈ R (T ).

Sj

The claim of the corollary now follows as a simple application of Freudenthal’s Theorem, [11, page 217].

j=1

with hα ↑ Pe. Let Λn = {α | hα ∈ Sn }

5 Applications

and qn = sup hα

In this section we consider the relevance of the results in Section 4 to classical and fuzzy probability theory. then qn ∈ Sn , as Sn is a Riesz subspace of E, see [1, Let ( fn ) be a sequence of random variables in a probpages 66 and 84]. Also qn ↑ Pe. Denote Pn = Pqn , then ability space (Ω, A , µ) with finite expectation. Let B be qn ≤ Pn e ≤ Pe and thus Pn ↑ P. As P is a tail band pro- a sub-σ-algebra of A and the sequence ( fn ) be indepenjection, Pe is in the tail of ( fn ) with respect to e, but dent with respect to each of the conditional probability \

measures µB , B ∈ B . Then by Corollary 4.5, the tail σ {e, f j , f j+1 , . . .} ⊂ h{e, fn+1 , fn+2 , . . .}i , algebra is a sub-σ-algebra of B . τ[( fn ), e] = j∈N We can similarly deduce a fuzzy Kolmogorov zeroone law as a consequence of Theorem 4.4 and Corollary and thus 4.5 by making the following identifications. Let (Ω, A ) Pe ∈ h{e, fn+1 , fn+2 , . . .}i , (4.1) be a measurable space, then a fuzzy random variable, [7], is a map f : Ω 7→ F(X) with for each n ∈ N. The independence of ( f j ) with respect {ω ∈ Ω | [ f (ω)]α ∩ B 6= φ} ∈ A , for all B open in X,(5.1) to T now gives that Pn e and Pe are independent since α∈Λn

where X is a Banach space and F(X) is the set of ν : X 7→ [0, 1] with να closed for α ∈ [0, 1], and να 6= φ for 0 < α ≤ 1 in which

Pn e ∈ h{e, f1 , . . . , fn }i and from (4.1)

να = {x ∈ X | ν(x) ≥ α}.

Pe ∈ h{e, fn+1 , fn+2 , . . .}i

(5.2)

This fuzzy structure can be embedded in the Riesz which have disjoint index sets, i.e. {1, . . . , n} and {n + space theory by taking X to be a topological space and 1, . . .}. Hence T Pn T P = T Pn P = T PT Pn ,

F(X) = {ν | ν : X 7→ R}

for all n ∈ N.

From the order continuity of T , taking the order limit with pointwise addition, scalar multiplication and paras n → ∞ in the above equation yields T PT P = T P, tial ordering making F(X) into a Riesz space. Let να be as defined in (5.2) for each ν ∈ F(X). Let (Ω, A ) making P self-independent with respect to T . be a measurable space and define the space of fuzzy Theorem 4.4 has the following notable corollary. measurable maps to be L(Ω, X), where f ∈ L(Ω, X) 5

if f : Ω 7→ F(X) and (5.1) holds. Then L(Ω, X) is a [10] P. Ter´an, ”An embedding theorem for convex fuzzy sets”, Fuzzy Sets and Systems, to appear. Dedekind complete Riesz space with weak order unit. Hence Theorem 4.4 and Corollaries 4.2 and 4.5 are applicable in L(Ω, X). We define the set of fuzzy ran- [11] A. C. Zaanen (1997) ”Introduction to Operator Theory in Riesz Space”, Springer Verlag. dom variables F(Ω, X) to be that subset of L(Ω, X) defined by f ∈ F(Ω, X) if f ∈ L(Ω, X) with fα (x) ∈ [0, 1]. As F(Ω, X) is a solid, Dedekind complete subset of L(Ω, X), Theorem 4.4 and Corollaries 4.2 and 4.5 apply in F(Ω, X). Ter´an in [10] gives an embedding of a class of fuzzy sets into a C(X) space and thus into a Riesz space. Hence our results apply and give rise to a Kolmogorov zero-one law for fuzzy sets.

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