Vol. LXXXI, 1 (2012), pp. 31–54

LEVY’S THEOREM AND STRONG CONVERGENCE OF MARTINGALES IN A DUAL SPACE

M. SAADOUNE

Abstract. We prove Levy’s Theorem for a new class of functions taking values from a dual space and we obtain almost sure strong convergence of martingales and mils satisfying various tightness conditions.

1. Introduction

This work is devoted to the study of strong convergence of martingales and mils
in the space L^{1}_{X}∗[X](Ω,F, P) of X-scalarly measurable functions f such that
ω → kf(ω)k is P-integrable, where (Ω,F, P) is a complete probability space,
X is a separable Banach space andX^{∗} is its topological dual without the Radon-
Nikodym Property. By contrast with the well known Chatterji result dealing
with strong convergence of relatively weakly compactL^{1}_{Y}(Ω,F, P)-bounded mar-
tingales, whereY is a Banach space, the case of the space L^{1}_{X}∗[X](Ω,F, P) con-
sidered here is unusual because the functions are no longer strongly measurable,
the dual space is not strongly separable. Our starting point of this study is to
characterize functions in L^{1}_{X}∗[X](Ω,F, P) whose associated regular martingales
almost surely strong converge, by introducing the notion ofσ-measurability. We
then proceed by stating our main results, which stipule that under various tightness
conditions,L^{1}_{X}∗[X](Ω,F, P)-bounded martingales and mils almost surely converge
with respect to the strong topology onX^{∗}. Further, we study the special case of
martingales in the subspace of L^{1}_{X}∗[X](F) of all Pettis-integrable functions that
satisfy a condition formulated in the manner of Marraffa [25]. For the weak star
convergence of martingales and mils taking values from a dual space, the reader is
referred to Fitzpatrick-Lewis [20] and the recent paper of Castaing-Ezzaki-Lavie-
Saadoune [7].

The paper is organized as follows. In Section 2 we set our notations and defini-
tions, and summarize needed results. In section 3 we present a weak compactness
result for uniformly integrableweak tight sequences in the spaceL^{1}_{X}∗[X](Ω,F, P)
as well as we give application to biting lemma. These results will be used in the

Received February 24. 2011; revised June 19, 2011.

2010Mathematics Subject Classification. Primary 60B11, 60B12, 60G48.

Key words and phrases. σ-measurable function; conditional expectation; martingale; mil;

Levy’s theorem; tightness; sequential weak upper limits; weak-star; weak and strong convergence.

M. SAADOUNE

next sections. In Section 4σ-measurable functions are presented and Levy’s theo-
rem for such functions is stated. In Section 5 we give our main martingale almost
surely strong convergence result (Theorem 5.1) accompanied by some important
Corollaries 5.1–5.3. A version of Theorem 5.1 for mils is provided at the end of this
section (Theorem 5.2). Finally, in Section 6 we discuss the special case of bounded
martingales inL^{1}_{X}∗[X](Ω,F, P) whose members are also Pettis integrable. It will
be shown that for such martingales it is possible to pass from convergence in a
very weak sense (see [25], [17], [4]) to strong convergence (Proposition 6.1).

2. Notations and Preliminaries

In the sequel,X is a separable Banach space and (x`)_{`≥1}is a fixed dense sequence
in the closed unit ballBX. We denote byX^{∗}the topological dual ofXand the dual
norm byk.k. The closed unit ball ofX^{∗}is denoted byBX^{∗}. Iftis a topology onX^{∗},
the spaceX^{∗}endowed withtis denoted byX_{t}^{∗}. Three topologies will be considered
onX^{∗}, namely the norm topologys^{∗}, the weak topologyw=σ(X^{∗}, X^{∗∗}) and the
weak-star topologyw^{∗}=σ(X^{∗}, X).

Let (C_{n})_{n≥1} be a sequence of subsets ofX^{∗}. The sequential weak upper limit
w−ls C_{n} of (C_{n}) is defined by

w−ls C_{n} ={x∈X^{∗}: x=w− lim

j→+∞x_{n}_{j}, x_{n}_{j} ∈C_{n}_{j}}

and thetopological weak upper limitw−LS C_{n} of (C_{n}) is denoted byw−LS C_{n}
and is defined by

w−LS Cn = \

n≥1

w−cl [

k≥n

Cn,

wherew−cl denotes the closed hull operation in the weak topology. The following inclusion

w−ls Cn⊆w−LS Cn

is easy to check. Conversely, if theCn are contained in a fixed weakly compact subset, then both sides coincide.

Let (Ω,F, P) be a complete probability space. A functionf: Ω→X^{∗}is said to
be X-scalarly F-measurable (or simply scalarlyF-measurable) if the real-valued
functionω→< x, f(ω)>is measurable with respect to (w.r.t.) theσ-field F for
allx∈X. We say also thatf is weak^{∗}-F-measurable. Recall that if f: Ω→X^{∗}
is a scalarlyF-measurable function such thathx, fi ∈L^{1}_{R}(F) for allx∈X, then
for eachA∈ F, there isx^{∗}∈X^{∗}such that

∀x∈X, hx, x^{∗}i=
Z

A

hx, fidP.

The vectorx^{∗} is called the weak^{∗} integral (or Gelfand integral) off overAand is
denoted simplyR

Af dP. We denote byL^{0}_{X}∗[X](F) (resp. L^{1}_{X}∗[X](F)) the space
of all (classes of) scalarly F-measurable functions (resp. scalarly F-measurable
functions f such that ω → kf(ω)k is P-integrable). By [14, Theorem VIII.5]

(actually, a consequence of it) (see also [3, Proposition 2.7]),L^{1}_{X}∗[X](F) endowed
with the normN1 defined by

N_{1}(f) :=

Z

Ω

kfkdP, f ∈L^{1}_{X}∗[X](F),

is a Banach space. For more properties of this space, we refer to [3] and [14].

Next, let (Fn)_{n≥1}be an increasing sequence of sub-σ-algebras ofF. We assume
without loss of generality that F is generated by ∪nFn. A function τ: Ω →
N∪ {+∞}is called a stopping time w.r.t. (Fn) if for eachn≥1, {τ =n} ∈ Fn.
The set of all bounded stopping times w.r.t. (Fn) is denotedT. Let (fn)_{n≥1} be a
sequence inL^{1}_{X}∗[X](F). If eachfn isFn-scalarly measurable, we say that (fn) is
adapted w.r.t. (Fn). Forτ ∈T and (fn) an adapted sequence w.r.t. (Fn) recall
that

fτ:=

max(τ)

X

k=min(τ)

fk1_{{τ=k}} and Fτ={A∈ F:A∩ {τ=k} ∈ Fk,∀k≥1}.

It is readily seen that f_{τ} is F_{τ}-scalarly measurable. Moreover, given a stopping
timeσ(not necessarily bounded), the following useful inclusion holds

{σ= +∞} ∩ F ⊂σ(∪nF_{σ∧n}),
(‡)

which is equivalent to

(‡)^{0} {σ= +∞} ∩ F_{m}⊂σ(∪_{n}F_{σ∧n}), for all m≥1,

whereσ∧nis the bounded stopping time defined byσ∧n(ω) := min(σ(ω), n) and
σ(∪nF_{σ∧n}) is the sub-σ-algebra ofF generated by∪nF_{σ∧n}. To verify (‡)^{0}, fixA
inFm and consider the sequence (fn) defined by fn := 1A ifn=m, 0 otherwise.

Then (fn) is adapted w.r.t (Fn) and it is easy to check the following equality
1_{{σ=+∞}}f_{σ∧m}= 1_{{σ=+∞}∩A}

with 1_{∅} = 0. As{σ= +∞} ∈σ(∪nF_{σ∧n}) (because{σ <+∞}=∪n{σ=n}and
{σ=n} ∈ F_{σ∧n}, for alln≥1), it follows that 1_{{σ=+∞}}f_{σ∧m}is measurable w.r.t.

σ(∪nF_{σ∧n}) and so is the function 1_{{σ=+∞}∩A}. Equivalently {σ = +∞} ∩A ∈
σ(∪nF_{σ∧n}). Thus{σ= +∞} ∩ Fm⊂σ(∪nF_{σ∧n}). Since this holds for allm≥1,
the inclusion (‡)^{0} follows.

Definition 2.1. An adapted sequence (fn)n≥1 in L^{1}_{X}∗[X](F) is a martingale
if

Z

A

fndP = Z

A

fn+1dP

for eachA∈ Fn and eachn≥1. EquivalentlyE^{F}^{n}(fn+1) =fn for eachn≥1.

E^{F}^{n}denotes the (Gelfand) conditional expectation w.r.t. Fn. It must be noted
that the conditional expectation of a Gelfand function inL^{1}_{X}∗[X](F) always exists,
(see [32, Proposition 7, p. 366] and [35, Theorem 3]).

M. SAADOUNE

Definition 2.2. An adapted sequence (fn)_{n≥1} in L^{1}_{X}∗[X](F) is a mil if for
everyε >0, there existspsuch that for eachn≥p, we have

P( sup

n≥q≥p

kfq−E^{F}^{q}f_{n}k> ε)< ε.

It is obvious that if (f_{n})_{n≥1}is a mil inL^{1}_{X}∗[X](F), then for everyxinB_{X}, the
sequence (hx, fni)n≥1 is a mil inL^{1}

R(F).

We end this section by recalling two concepts of tightness which permit us to
pass from weak star to strong convergence. For this purpose, letC = cwk(X_{w}^{∗})
or R(X_{w}^{∗}), where cwk(X_{w}^{∗}) (resp. R(X_{w}^{∗})) denotes the space of all nonempty
σ(X^{∗}, X^{∗∗})-compact convex subsets of X_{w}^{∗} (resp. closed convex subsets of X_{w}^{∗}
such that their intersections with any closed ball are weakly compact). AC-valued
multifunction Γ : Ω⇒X^{∗} isF-measurable if its graphGr(Γ) defined by

Gr(Γ) :={(ω, x^{∗})∈Ω×X^{∗}:x^{∗}∈Γ(ω)}

belongs toF ⊗ B(X_{w}^{∗}∗).

Definition 2.3. A sequence (fn) in L^{0}_{X}∗[X](F) is C-tight if for every ε > 0,
there is aC-valuedF-measurable multifunction Γε: Ω⇒X^{∗} such that

inf

n P({ω∈Ω :f_{n}(ω)∈Γ_{ε}(ω)})≥1−ε.

In view of the completeness hypothesis on the probability space (Ω,F, P), the
measurability of the set{ω∈Ω :fn(ω)∈Γε(ω)} is a consequence of the classical
Projection Theorem [14, Theorem III.23] sinceX_{w}^{∗}∗ is a Suslin space and Γε has
its graph inF ⊗ B(X_{w}^{∗}∗) (see [8, p. 171–172] and also [6, 11]).

Now let us introduce a weaker notion of tightness, namelyS(C)-tightness. It is a dual version of a similar notion in [6] dealing with primal spaceX.

Definition 2.4. A sequence (fn) inL^{0}_{X}∗[X](F) isS(C)-tight if there exists a
C-valuedF-measurable multifunction Γ : Ω⇒X^{∗} such that for almost all ω∈Ω,
one has

fn(ω)∈Γ(ω) for infinitely many indices n.

(*)

The following two results reformulate [6, Proposition 3.3] for sequences of mea- surable functions with values in a dual space.

Proposition 2.1. Let (fn) be an R(X_{w}^{∗})-tight sequence. If it is bounded in
L^{1}_{X}∗[X](F), then it is also cwk(X_{w}^{∗})-tight.

Proof. Letε >0. By theR(X_{w}^{∗})-tightness assumption, there exists a F-meas-
urableR(X_{w}^{∗})-valued multifunction Γε: Ω⇒X^{∗} such that

infn P({ω∈Ω :f_{n}(ω)∈Γ_{ε}(ω)})≥1−ε.

(2.1)

On the other hand, since (kfnk) is bounded inL^{1}

R^{+}(F), one can find rε>0 such
that

sup

n

P({kfnk> r_{ε}})≤ε.

(2.2)

For eachn≥1, put

An,ε:={ω∈Ω :fn(ω)∈Γε(ω)∩B(0, rε)}

and let us consider the multifunction ∆_{ε}defined on Ω by

∆ε:=s^{∗}-cl co [

n≥1

{1A_{n,ε}fn}
.

The values of multifunction ∆εarecwk(X_{w}^{∗})-valued, because ∆ε(ω)⊂s^{∗}-cl co({0}

∪[Γε(ω)∩B(0, rε)]) and Γε(ω)∈ R(X_{w}^{∗}), for allω. Therefore, ∆εisF-measurable
(see [6], [10]). Finally, using (2.1), (2.2) and the following inclusions

A_{n,ε}⊆ {ω∈Ω :f_{n}(ω)∈∆_{ε}(ω)}, n≥1,
we get

P({ω∈Ω :f_{n}(ω)∈∆_{ε}(ω)})>1−2ε for alln.

Proposition 2.2. Every C-tight sequence is S(C)-tight.

Proof. Let (f_{n}) be a C-tight sequence in L^{0}_{X}∗[X](F) and consider ε_{q} := ^{1}_{q},
q ≥1. By the C-tightness assumption, there is a F-measurable C-valued multi-
function Γ_{ε}_{q} : Ω⇒X^{∗} denoted simply Γ_{q} such that

infn P(An,q)≥1−εq, (2.3)

where

An,q:={ω∈Ω :fn(ω)∈Γq(ω)}.

Now, we define the sequence (Ω_{q})_{q≥1} by
Ωq = lim sup

n→+∞

An,q

and the multifunction Γ on Ω by

Γ = 1Ω^{0}_{1}Γ1+X

q≥2

1Ω^{0}_{q}Γq,

where Ω^{0}_{1}= Ω1 and Ω^{0}_{q} = Ωq\ ∪i<qΩi for allq >1. Then inequality (2.3) implies
P(Ωq) = lim

n→∞P [

m≥n

Am,q

≥1−εq →1.

Further, for eachω∈Ωq, one has

ω∈A_{n,q} ={ω∈Ω :f_{n}(ω)∈Γ(ω)} for infinitely many indices n.

This proves theS(C)-tightness.

Remark 2.5. By the Eberlein-Smulian theorem, the following implication
(f_{n}) S(cwk(X_{w}^{∗}))-tight⇒w−ls f_{n}6=∅ a.s.

holds true. Conversely, ifw−ls fn 6=∅a.s. then the condition (*) in Definition 2.4 is satisfied, but the multifunctionC may fail to beF-measurable.

M. SAADOUNE

Actually, in all results involving theS(C)-tightness condition, the measurability of the multifunction Γ is not essential.

3. weak compactness in the spaceL^{1}_{X}∗[X](F)

We recall first the following weak compactness result in the spaceL^{1}_{X}∗[X](F) due
to Benabdellah and Castaing [3].

Proposition 3.1. ([3, Proposition 4.1]) Suppose that (fn)_{n≥1} is a uniformly
integrable sequence in L^{1}_{X}∗[X](F) and Γ is a cw(X_{w}^{∗})-valued multifunction such
that

fn(ω)∈Γ(ω) a.s. for all n≥1,
then(fn)is relatively weakly compact in L^{1}_{X}∗[X](F).

Proceeding as in the primal case (see [5], [1], [30]), it is possible to extend this
result to uniformly integrableR(X_{w}^{∗})-tight sequences inL^{1}_{X}∗[X](F)

Proposition 3.2. Suppose that(f_{n})_{n≥1} is a uniformly integrableR(X_{w}^{∗})-tight
sequence inL^{1}_{X}∗[X](F). Then (f_{n})is relatively weakly compact in L^{1}_{X}∗[X](F).

Proof. By Proposition 2.1, (f_{n}) is cwk(X_{w}^{∗})-tight since it is bounded and
R(X_{w}^{∗})-tight. Consequently, for everyq≥1, there is aF-measurablecwk(X_{w}^{∗})-va-
lued multifunction Γ1

q: Ω⇒X^{∗}, denoted simply Γ_{q}, such that
infn P(An,q)≥1−1

q, where

An,q:={ω∈Ω :fn(ω)∈Γq(ω)}.

Now, for eachq≥1, we consider the sequence (f_{n,q}) defined by
fn,q = 1A_{n,q}fn n≥1.

By Proposition 3.1, the sequence (fn,q) is relatively weakly compact inL^{1}_{X}∗[X](F)
since it isL^{1}_{X}∗[X](F)-bounded andfn,q(ω) belongs to thew-compact set Γ(ω) for
allω∈Ω and alln,q≥1. Furthermore, we have the following estimation

sup

n

Z

Ω

kfn−fn,qkdP ≤sup

n

Z

Ω\An,q

kfnkdP

for allq≥1. As (f_{n}) is uniformly integrable and inf_{n}P(A_{n,q})≥1−^{1}_{q}, we get

q→∞lim sup

n

Z

Ω\An,q

kfnkdP= 0.

Hence

q→∞lim sup

n

Z

Ω

kf_{n}−f_{n,q}kdP= 0.

Consequently, by Grothendieck’s weak relative compactness lemma ([22, Chap. 5,
4, n^{◦}1]), the sequence (fn) is relatively weakly compact inL^{1}_{X}∗[X](F).

Now, we provide the following version of the biting lemma in the space
L^{1}_{X}∗[X](F). See [13] for other related results involving a weaker mode of con-
vergence; see also [9] dealing with the primal case.

Proposition 3.3. Let (fn)be a boundedR(X_{w}^{∗})-tight sequence inL^{1}_{X}∗[X](F).

Then there exist a subsequence(f_{n}^{0}) of (fn), a functionf_{∞}∈L^{1}_{X}∗[X](F) and an
increasing sequence (Bp) of measurable sets with lim_{p→∞}P(Bp) = 1 such that
(1B_{p}f_{n}^{0}) converges to1B_{p}f_{∞} in the weak topology ofL^{1}_{X}∗[X](F)for allp≥1.

Proof. In view of the biting lemma (see [21], [33] [31]), there exist an increasing
sequence (Bp) of measurable sets with lim_{p→∞}P(Bp) = 1 and a subsequence (f_{n}^{0})
of (fn) such that for allp≥1, the sequence (1B_{p}f_{n}^{0}) is uniformly integrable. It is
alsoR(X_{w}^{∗})-tight. Consequently, by Proposition 3.2, for eachp≥1, (1B_{p}f_{n}^{0}) is rel-
atively weakly compact inL^{1}_{X}∗[X](F). By applying the Eberlein-Smulian theorem
via a standard diagonal procedure, we provide a subsequence of (f_{n}^{0}), not relabeled,
such that for eachp≥1, (1_{B}_{p}f_{n}^{0}) converges to a functionf_{∞,p}∈L^{1}_{X}∗[X](F) in the
weak topology ofL^{1}_{X}∗[X](F), also denotedσ(L^{1}_{X}∗[X](F),(L^{1}_{X}∗[X](F))^{0}). Finally,
define

f_{∞}:=

p=∞

X

p=1

1C_{p}f_{∞,p},
where

C1:=B1 and Cp:=Bp\ ∪i<pBi for p >1.

It is not difficult to verify that (1B_{p}f_{n}^{0}) converges to 1B_{p}f_{∞}in the weak topology of
L^{1}_{X}∗[X](F). Since the normN1(.) ofL^{1}_{X}∗[X](F) isσ(L^{1}_{X}∗[X](F),(L^{1}_{X}∗[X](F))^{0})-
lower semi-continuous, we have

Z

Bp

kf_{∞}kdP ≤lim inf

n→∞

Z

Bp

kf_{n}^{0}kdP ≤sup

n

Z

Ω

kfnkdP <∞ for all p≥1.

As lim_{p→∞}P(B_{p}) = 1, we deduce thatkf∞k ∈L^{1}

R(F). This completes the proof

of Proposition 3.3.

As a consequence of Proposition 3.3 and Mazur theorem we get the following corollary.

Corollary 3.1. Let (f_{n}) be a bounded R(X_{w}^{∗})-tight sequence in L^{1}_{X}∗[X](F).

Then there exist a sequence (g_{n}) with g_{n} ∈ co{f_{i} : i ≥n} and a function f_{∞} ∈
L^{1}_{X}∗[X](F) such that

(gn)s^{∗}-converges to f_{∞} a.s.

Proof. By the assumptions and Proposition 3.3, there exist a subsequence (f_{n}^{0})
of (fn), a functionf_{∞}∈L^{1}_{X}∗[X](F) and increasing sequence (Bp) of measurable
sets with lim_{p→∞}P(Bp) = 1 such that for allp≥1, (1B_{p}f_{n}^{0}) converges to 1B_{p}f_{∞}
in the weak topology ofL^{1}_{X}∗[X](F). So, appealing to a diagonal procedure based
on successively applying Mazur’s theorem (see [10, Lemma 3.1]), one can show the
existence of a sequence (gn) of convex combinations of (f_{n}^{0}), such that for allp≥1,
(1B_{p}gn) s^{∗}-converges almost surely to 1B_{p}f∞ and also strongly in L^{1}_{X}∗[X](F).

Since limp→∞P(Bp) = 1, (gn)s^{∗}-converges almost surely tof∞.

M. SAADOUNE

4. Levy’s theorem inL^{1}_{X}∗[X](F)

In this section, we present a new class of functions inL^{1}_{X}∗[X](F) whose associated
regular martingales almost surely converge with respect to the strong topology of
X^{∗}.

Definition 4.1. A function f in L^{0}_{X}∗[X](F) is said to be σ-measurable, if
there exists an adapted sequence (Γ_{n})_{n≥1} (that is, for each integer n≥1, Γ_{n} is
Fn-measurable) ofR(X_{w}^{∗})-valued multifunctions such that f(ω)∈s^{∗}-cl co(∪nΓn)
a.s.

Remark 4.2. The sequence (Γn) given in this definition can be assumed to be adapted w.r.t. a subsequence of (Fn).

Remark 4.3. As a special case note thatevery strongly measurable function
f: Ω → X^{∗} is σ-measurable. Indeed, if (ξn)_{n≥1} is a sequence of measurable
functions assuming a finite number of values and which norm converges a.s. tof,
thenf(ω)∈s^{∗}-cl(∪_{n≥1}ξn(Ω)) a.s., (Γn:=ξn(Ω)).

Proposition 4.1. Let f ∈ L^{0}_{X}∗[X](F) and suppose there exists a sequence
(Γn)_{n≥1}of R(X_{w}^{∗})-valued multifunctions which is adapted w.r.t. a subsequence of
(Fn) such thatf(ω)∈s^{∗}-cl co w-LSΓn a.s., thenf isσ-measurable.

Proof. Indeed, since w-LSΓn:= \

k≥1

w-cl [

n≥k

Γn

⊂ \

k≥1

s^{∗}-cl co [

n≥k

Γn

⊂s^{∗}-cl co [

n≥1

Γn , we have

s^{∗}-cl cow-LSΓn ⊂s^{∗}-cl co [

n≥1

Γn

.

In particular, we have the following result.

Corollary 4.1.Letf∈L^{0}_{X}∗[X](F). If there exists a sequence(fn)inL^{0}_{X}∗[X](F),
adapted w.r.t. a subsequence of (Fn) which weak converges a.s. to f, then f is
σ-measurable.

The following proposition will be useful in this work.

Proposition 4.2. Let (f_{n})_{n≥1} be an adapted S(cwk(X_{w}^{∗}))-tight sequence in
L^{0}_{X}∗[X](F) andf_{∞} a function inL^{0}_{X}∗[X](F)such that

n→∞limhx`, fni=hx`, f∞i a.s. for all `.

Thenf_{∞} isσ-measurable.

Proof. S(cwk(X_{w}^{∗}))-tightness and Remark 2.5 imply
w-ls f_{n}6=∅ a.s.

Since limn→∞hx`, fni=hx`, f∞i, it is easy to prove that
w-ls fn={f_{∞}} a.s.

Thusf_{∞} isσ-measurable, in view of Proposition 4.1
There are two significant variants of Proposition 4.2. involving the
R(X_{w}^{∗})-tightness condition. The first one is essentially based on Proposition 3.2.

Proposition 4.3. Let (fn)n≥1 be a uniformly integrable R(X_{w}^{∗})-tight adapted
sequence inL^{1}_{X}∗[X](F)andf∞ a function in L^{1}_{X}∗[X](F). Suppose there exists a
sequence(g_{n})inL^{1}_{X}∗[X](F)withg_{n} ∈co{fi:i≥n} such that

n→∞limhx`, gni=hx`, f_{∞}i a.s. for all `.

Thenf_{∞} isσ-measurable.

Proof. Let (gn) be given as in the proposition. By Proposition 3.2 and Krein-
Smulian theorem, the convex hull of the set {fn : n ≥ 1} is relatively weakly
compact in L^{1}_{X}∗[X](F); hence (gn) is relatively weakly compact in L^{1}_{X}∗[X](F).

Consequently, by the Eberlein Smulian theorem, there exists a subsequence of
(g_{n}), not relabeled, such that for each p≥1, (g_{n}) converges to a functionf_{∞}^{0} ∈
L^{1}_{X}∗[X](F) in the weak topology of L^{1}_{X}∗[X](F). So, invoking Mazur’s theorem
it can be shown the existence of a sequence of convex combinations of (g_{n}), still
denoted in the same manner such that (g_{n})s^{∗}-converges almost surely tof_{∞}^{0} . As
lim_{n→∞}hx_{`}, g_{n}i = hx_{`}, f_{∞}ia.s. for all `, we get f_{∞} = f_{∞}^{0} a.s. Therefore, since
(g_{n}) is adapted w.r.t. a subsequence of (F_{n}), it follows thatf_{∞}is σ-measurable.

The second variant is a consequence of the proof of Corollary 3.1.

Proposition 4.4. Let (fn)n≥1 be a bounded R(X_{w}^{∗})-tight adapted sequence in
L^{1}_{X}∗[X](F) and f∞ a function in L^{1}_{X}∗[X](F) such that the following condition
holds.

For any subsequence(f_{n}^{0}) of (f_{n}), there is a sequence (g_{n})in L^{1}_{X}∗[X](F)with
g_{n}∈co{f_{i}^{0}:i≥n} such that

n→∞limhx`, gni=hx`, f_{∞}i a.s. for all `.

Thenf_{∞} isσ-measurable.

Now our main result comes and shows that a regular martingale associated to
aσ-measurable function inL^{1}_{X}∗[X](F) norm converges a.s.

Proposition 4.5. Let f be a function inL^{1}_{X}∗[X](F). Then the following two
statements are equivalent:

(a) (E^{F}^{n}(f))s^{∗}-converges a.s.to f;
(b) f is σ-measurable.

Proof. Step 1. The implication (a)⇒ (b) is trivial. Conversely, suppose that
f isσ-measurable. Then there exists an adapted sequence (Γn) ofR(X_{w}^{∗})-valued
multifunctions such that

f(ω)∈s^{∗}-cl co [

n

Γ_{n}(ω)
a.s.

(4.1)

M. SAADOUNE

Without loss of generality, we may suppose that 0∈Γn(ω), for allω ∈Ω and all
n≥1. For eachn, p≥1, define the multifunction Γ^{p}_{n} by

Γ^{p}_{n} := Γn∩BX^{∗}(0, p).

Since this multifunction isFn-measurable, namelyGr(Γ^{p}_{n})∈ Fn⊗B(X_{w}^{∗}∗) andX_{w}^{∗}∗

is a Suslin space, invoking [14, Theorem III.22], one can find a sequence (σ_{n,i}^{p} )i≥1

of scalarlyFn-measurable selectors of Γ^{p}_{n} that are alsoL^{1}_{X}∗[X](F)-integrable (be-
cause the multifunctions Γ^{p}_{n} are integrably bounded) such that for everyω∈Ω,

w^{∗}−cl(Γ^{p}_{n}(ω)) =w^{∗}−cl({σ_{n,i}^{p} (ω)}_{i≥1}).

Equivalently

Γ^{p}_{n}(ω) =w−cl({σ_{n,i}^{p} (ω)}_{i≥1}),
since Γ^{p}_{n} isw-compact valued. So

Γ^{p}_{n}(ω)⊂w−cl co({σ_{n,i}^{p} (ω)}i≥1) =s^{∗}-cl co({σ_{n,i}^{p} (ω)}i≥1).

(4.2)

Let (sm)_{m≥1} be the sequence of all linear combinations with rational coefficients
ofσ_{n,i}^{p} , (n, p, i≥1). It is easy to check that

s^{∗}-cl co({σ^{p}_{n,i}(ω)}_{n,i,p≥1})⊂s^{∗}−cl({sm(ω)}_{m≥1}).

Combining this with (4.2) we get
s^{∗}-cl co [

n

Γ_{n}(ω)

=s^{∗}-cl co [

n

[

p

Γ^{p}_{n}(ω)

⊂s^{∗}−cl({s_{m}(ω)}_{m≥1}),
whence, by (4.1)

f(ω)∈s^{∗}−cl({sm(ω)}_{m≥1}) a.s.

(4.3)

Now, for eachq≥1, let us define the sets
B_{m}^{q} :=

ω∈Ω : kf(ω)−s_{m}(ω)k< 1
q

(m≥1),
Ω^{q}_{1}:=B_{1}^{q}, Ω^{q}_{m}:=B_{m}^{q} \ [

i<m

B_{i}^{q} form >1
and the function

fq :=

+∞

X

m=1

1_{Ω}^{q}_{m}sm.

Since the functions ω → kf(ω)−sm(ω)k are F-measurable, B^{q}_{m} ∈ F, for all
m≥1, and then eachfq is scalarly F-measurable. Further, from (4.3) it follows
that ∪mB^{q}_{m} = Ω a.s., so that (Ω^{q}_{m})m constitutes a sequence of pairwise disjoint
members ofF which satisfies∪mΩ^{q}_{m}= Ω a.s., and so we have

kf(ω)−f_{q}(ω)k ≤ 1

q for almost all ω∈Ω.

(4.4)

Next, we claim that

n→∞lim kE^{F}^{n}(f)−fk= 0 a.s.

First, observe that by construction of the sm’s, we can find a strictly increasing
sequence (pm) of positive integers such that (sm) is adapted w.r.t. (Fp_{m}). Now,
letk≥1 be a fixed integer. For each n≥pk, one has

E^{F}^{n}(1_{∪}m=k

m=1B_{m}^{q}fq) =E^{F}^{n}(1_{∪}m=k

m=1Ω^{q}_{m}fq) =E^{F}^{n}

m=k

X

m=1

1_{Ω}^{q}_{m}sm=

m=k

X

m=1

(E^{F}^{n}1_{Ω}^{q}_{m})sm,
whence by the classical Levy theorem

n→∞lim E^{F}^{n}(1_{∪}m=k

m=1B_{m}^{q}f_{q}) =

m=k

X

m=1

1_{Ω}^{q}_{m}s_{m}= 1_{∪}m=k

m=1B_{m}^{q}f_{q} a.s.

(4.5)

w.r.t. the norm topology ofX^{∗}. On the other hand, from (4.4) we deduce the
following estimation

kE^{F}^{n}(1_{∪}m=k

m=1B_{m}^{q}f)−1_{∪}m=k

m=1B^{q}_{m}fk ≤ kE^{F}^{n}(1_{∪}m=k

m=1B_{m}^{q}f)−E^{F}^{n}(1_{∪}m=k
m=1B_{m}^{q}fq)k
+kE^{F}^{n}(1_{∪}m=k

m=1B^{q}_{m}f_{q})−1_{∪}m=k
m=1B_{m}^{q}f_{q}k
+k1_{∪}m=k

m=1B_{m}^{q}f(ω)−1_{∪}m=k

m=1B_{m}^{q}f_{q}(ω)k

≤ kE^{F}^{n}(1_{∪}m=k

m=1Bm^{q}fq)−1_{∪}m=k

m=1B^{q}mfqk+2
q,
which leads to

kE^{F}^{n}(f)−fk ≤ kE^{F}^{n}(1_{∪}m=k

m=1B^{q}_{m}f)−1_{∪}m=k
m=1B^{q}_{m}fk
+kE^{F}^{n}(1_{Ω\∪}i=k

m=1B_{m}^{q}f)−1_{Ω\∪}m=k
m=1B_{m}^{q}fk

≤ kE^{F}^{n}(1_{∪}m=k

m=1B^{q}_{m}fq)−1_{∪}m=k

m=1B_{m}^{q}fqk+2
q
+E^{F}^{n}(1_{Ω\∪}m=k

m=1B_{m}^{q}kfk) + 1_{Ω\∪}m=k
m=1B^{q}_{m}kfk.

Consequently, from (4.5) and the classical Levy Theorem (kfk being in L^{1}

R(F)), it follows that

lim sup

n→∞

kE^{F}^{n}(f)−fk ≤2

1_{Ω\∪}m=k

m=1Bm^{q}kfk+1
q

,

a.s. for allk≥1 and allq≥1. SinceP(∪mB_{m}^{q} ) = 1, by passing to the limit when
k→ ∞andq → ∞, respectively, we get the desired conclusion, and the proof is

finished.

5. strong convergence of martingales inL^{1}_{X}∗[X](F)

The main result of this section asserts that under theS(R(X_{w}^{∗}))-tightness condi-
tion every bounded martingale inL^{1}_{X}∗[X](F) norm converges a.s. We begin with
the following decomposition result for martingales which is borrowed from [7]. For
the convenience of the reader we give a detailed proof.

Proposition 5.1. Let (f_{n})_{n≥1} be a bounded martingale inL^{1}_{X}∗[X](F). Then
there existsf_{∞}∈L^{1}_{X}∗[X](F)such that

n→∞lim kfn−E^{F}^{n}f_{∞}k= 0 a.s and,

M. SAADOUNE

(fn) w^{∗}-converges to f_{∞} a.s.

Proof. As (f_{n}) is a bounded martingale inL^{1}_{X}∗[X](F) for eachx∈X, (hx, fni)
is a bounded real martingale inL^{1}

R(F), hence it converges a.s. to a function r_{x}∈
L^{1}

R(F) for every x∈X. By using [11, Theorem 6.1(4)], we provide an increasing
sequence (A_{p})_{p≥1} inF with lim_{p→∞}P(A_{p}) = 1, a functionf_{∞}∈L^{1}_{X}∗[X](F) and
a subsequence (f_{n}^{0})_{n≥1} of (f_{n}) such that

n→∞lim Z

Ap

hh, f_{n}^{0}idP =
Z

Ap

hh, f_{∞}idP

for allp≥1 and allh∈L^{∞}_{X}(F). So by identifying the limit, we getr_{x}=hx, f_{∞}i
a.s. Hence

n→∞limhx, fni=hx, f_{∞}i, a.s. for all x∈X
(5.1)

and then in view of the classical Levy’s theorem

n→∞lim[hx, fni − hx, E^{F}^{n}(f_{∞})i] = 0 a.s. for all x∈X.

Furthermore,{(hx`, f_{n}i − hx`, E^{F}^{n}(f_{∞})i)n≥1:`≥1}is a countable family of real-
valuedL^{1}

R(F)-bounded martingales, thus invoking [28, Lemma V.2.9], we see that

n→∞lim kf_{n}−E^{F}^{n}f_{∞}k= lim

n→∞sup

`≥1

[hx_{`}, f_{n}i − hx_{`}, E^{F}^{n}(f_{∞})i)]

= sup

`≥1

n→∞lim[hx`, fni − hx`, E^{F}^{n}(f∞)i] = 0.

(5.2)

Since

sup

n

kE^{F}^{n}(f_{∞})k ≤sup

n

E^{F}^{n}kf_{∞}k<∞,
equation (5.2) entails

sup

n

kf_{n}k<∞ a.s.

Invoking the separability ofX and (5.1), we get
(fn) w^{∗}-converges to f_{∞} a.s.,

by a routine argument. This completes the proof.

Propositions 4.5 and 5.1 together allow us to pass from weak star convergence to strong convergence of martingales.

Theorem 5.1. Let (fn)n≥1 be a bounded martingale in L^{1}_{X}∗[X](F)satisfying
the following condition.

There exists aS(R(X_{w}^{∗}))-tight sequence(g_{n})in L^{1}_{X}∗[X](F)
(T)

with gn ∈co{fi:i≥n}.

Then there existsf_{∞}∈L^{1}_{X}∗[X](F)such that

(fn)s^{∗}-converges tof_{∞} a.s.

Proof. Let (gn) be as in condition (T). By Proposition 5.1, there existsf_{∞}∈
L^{1}_{X}∗[X](F) such that

kfn−E^{F}^{n}(f_{∞})k →0 a.s.

(a)

(fn) w^{∗}-converges to f_{∞} a.s.

(b)

By (b), (f_{n}) is pointwise bounded a.s., and so is the sequence (g_{n}). Consequently,
(g_{n}) is S(cwk(X_{w}^{∗}))-tight, since it is S(R(X_{w}^{∗}))-tight (by (T)). Furthermore, we
have

(g_{n})w^{∗}-converges to f_{∞} a.s.

Therefore, noting that (gn) is adapted w.r.t. a subsequence ofFn, we conclude
that f∞ isσ-measurable in view of Proposition 4.2. In turn, by Proposition 4.5,
this ensures the a.s. s^{∗}-convergence of E^{F}^{n}(f∞) to f∞. Coming back to (a), we

get the desired conclusion.

An alternative proof of Theorem 5.1 via a standard stopping time argument is also available. We want to emphasize that some of the arguments used in this proof will be helpful in the next section.

Second proof. Reasoning as at the beginning of the proof of Proposition 5.1 we
find a functionf_{∞}∈L^{1}_{X}∗[X](F) such that

n→∞limhx, fn(ω)i=hx, f∞(ω)i a.s. for all x∈X.

(5.3)

1) Suppose that sup_{n}kfnk ∈L^{1}_{R}(F). Then equation (5.3) implies

n→∞lim Z

A

hx, fnidP = Z

A

hx, f∞idP

for allx∈X and for allA∈ F. Since (fn) is a martingale, it follows that Z

A

hx, fmidP = lim

n→∞,n≥m

Z

A

hx, fnidP

= Z

A

hx, f∞idP= Z

A

hx, E^{F}^{m}(f∞)idP
for allx∈X,m≥1 andA∈ Fm. Hence

f_{m}=E^{F}^{m}(f_{∞}) a.s. for all m≥1,

by the separability ofX. On the other hand, the sequence (gn) appearing in the
condition (T) above is S(cwk(X_{w}^{∗}))-tight, since it isS(R(X_{w}^{∗}))-tight and point-
wise-bounded almost surely in view of the inequality

sup

n≥1

kg_{n}(ω)k ≤sup

n≥1

kf_{n}(ω)k<∞ a.s.

Further, from (5.3) it follows

n→∞limhx, gni=hx, f_{∞}i a.s.,

for every x ∈ X. Taking into account Proposition 4.2, it follows that f_{∞} is
σ-measurable.Therefore, by Proposition 4.5, (fn)s^{∗}-converges a.s. tof_{∞}.

M. SAADOUNE

2) The case sup_{n}R

ΩkfnkdP <∞. For eacht >0, define the following well known stopping time

σt(ω) =

( n if kfi(ω)k ≤t, for i= 1, . . . , n−1 and kfn(ω)k ≥t, +∞ if kfi(ω)k ≤t, for all i.

Then, following the same lines as those of the L^{1}_{E}(F) case ([15], [19]) we show
that:

(i) (fσt∧n,Fσt∧n) is aL^{1}_{X}∗[X](F)-bounded martingale.

(ii) The functionω→sup_{n}kfσt∧n(ω)kis integrable.

(iii) P(A_{t}:={ω:σ_{t}(ω) =∞})→1 ast→ ∞.

Moreover, using (5.3) it is not difficult to check that

n→∞limhx, f_{σ}_{t}_{∧n}(ω)i=hx, f_{∞}^{t} (ω)i, a.s.

(5.4)

for everyx∈X, where

f_{∞}^{t} (ω) :=

f_{∞}(ω) ifω∈At,
f_{σ}_{t}_{(ω)}(ω) otherwise.

By (5.4), it is clear thatf_{∞}^{t} is scalarlyF-measurable. Furthermore, one has
kf_{∞}^{t} k ≤lim inf

n→+∞kfσ_{t}∧nk a.s.

which in view of (i) and Fatou’s lemma (or (ii)) shows that kf_{∞}^{t} k is integrable.

Thusf_{∞}^{t} ∈L^{1}_{X}∗[X](F).

Now, writing eachg_{n} in the form
gn=

kn

X

i=n

µ^{i}_{n}fi with 0≤µ^{i}_{n}≤1 and

kn

X

i=n

µ^{i}_{n}= 1,
we define

g_{n}^{t}(ω) :=

k_{n}

X

i=n

µ^{i}_{n}fσ_{t}∧n(ω), (t >0).

Observing that
g^{t}_{n}(ω) =

gn(ω) if ω∈At,

f_{σ}_{t}_{(ω)}(ω) otherwise for all n≥σt(ω),

we conclude that (g_{n}^{t}(ω)) isS(R(X_{w}^{∗}))-tight and equation (5.4) entails the follow-
ing convergence

n→∞limhx, g_{n}^{t}(ω)i=hx, f_{∞}^{t} (ω)i, a.s.

for everyx∈X. Consequently, by (i), (ii), (5.4) and the first part of the proof, it
follows that (fσ_{t}∧n)s^{∗}-converges a.s. to f_{∞}^{t} . Since (fσ_{t}∧n) andf_{∞}^{t} respectively,
coincide with (fn) and f∞ onAt and P(At)→1 when t→ ∞ (in view of (iii)),

we deduce that (fn)s^{∗}-converges a.s. tof∞.

Now here are some important corollaries.

Corollary 5.1. Let(fn)_{n≥1} be a bounded martingale inL^{1}_{X}∗[X](F)satisfying
the following condition

There exists aR(X_{w}^{∗})-tight sequence(gn) with gn ∈co{fi:i≥n}.

(T^{+})

Then there existsf∞∈L^{1}_{X}∗[X](F)such that

(fn) s^{∗}-converges a.s. to f∞.

Proof. In view of Proposition 2.2, (T^{+}) implies (T). This implication is also a

consequence of Corollary 3.1.

As a special case of this corollary we obtain the following extension of Chatterji
result [16] (see also [19, Corollary II.3.1.7]) to the spaceL^{1}_{X}∗[X](F).

Corollary 5.2. Let (fn)_{n≥1} be a bounded martingale in L^{1}_{X}∗[X](F). Suppose
there exists acwk(X_{w}^{∗})-valued multifunction K such that

fn(ω)∈K(ω) for all n≥1.

Then there existsf∞∈L^{1}_{X}∗[X](F)such that(fn)s^{∗}-converges a.s. tof∞.
Corollary 5.3. Let (fn)_{n≥1} be a bounded martingale in L^{1}_{X}∗[X](F) and let
f_{∞}∈L^{1}_{X}∗[X](F)be such that

n→∞limhx`, fn(ω)i=hx`, f∞(ω)ia.s. for all `≥1.

(?)

Then the following statements are equivalent
(1) (f_{n}) s^{∗}-converges tof_{∞} a.s.

(2) There exists a sequence(g_{n})withg_{n} ∈co{f_{i}:i≥n}which a.s.w-converges
tof_{∞}.

(3) f_{∞} isσ-measurable.

Proof. The implication (1) ⇒ (2) is obvious, whereas (2) ⇒ (3) follows from Corollary 4.1.

(3)⇒(1): A close look at the first proof of Theorem 5.1 reveals that the condition

(T) may be replaced with (?) and (3).

It is worth to give the following variant of Proposition 5.1–Theorem 5.1.

Proposition 5.2. Let (f_{n})_{n≥1} be a martingale in L^{1}_{X}∗[X](F) satisfying the
following two conditions:

(C1) For each`≥1, there exists a sequence(hn)with hn∈co{fi :i≥n} such that (hx`, hni)is uniformly integrable.

(C2) lim infn→∞kfnk ∈L^{1}_{R}(F)

Then there existsf_{∞}∈L^{1}_{X}∗[X](F)such that

fn =E^{F}^{n}(f_{∞})for all n≥1 a.s. and
(f_{n}) w^{∗}-converges to f_{∞} a.s.

Furthermore, if the condition (T) is satisfied, then
(fn)s^{∗}-converges tof_{∞} a.s.

M. SAADOUNE

Proof. Let`≥1 be fixed and let (hn) be the sequence associated to`according
with (C1). As the sequence (hx`, hni) is uniformly integrable, there exist a subse-
quence (hn_{k}) of (hn) (possibly depending upon`) and a functionϕ`∈L^{1}_{R}(F) such
that

k→∞lim Z

A

hx`, hn_{k}idP =
Z

A

ϕ`dP

for everyA ∈ F. Sinceh_{n} ∈co{f_{i} :i≥n} and (hx_{`}, f_{n}i)_{n} is a martingale, it is
easy to check that

Z

A

hx`, hn_{k}idP =
Z

A

hx`, fmidP for allk≥mandA∈ Fm. Therefore

Z

A

hx`, fmidP = Z

A

ϕ`dP for all A∈ Fm

which is equivalent to

hx`, fmi=E^{F}^{m}(ϕ`) a.s.

(5.5)

This holds for all`≥1 andm≥1. Using the classical Levy’s theorem, we get

n→+∞lim hx`, fni=ϕ` a.s. for all `≥1.

(5.6)

On the other hand, by (C_{2}) and the cluster point approximation theorem [2,
Theorem 1]), (see also [18]), there exists an increasing sequence (τ_{n}) in T with
τ_{n} ≥nfor alln, such that

n→∞lim kfτnk= lim inf

n→∞ kfnk a.s.

Then, for each ω outside a negligible set N, the sequence (fτ_{n}(ω)) is bounded
in X^{∗}; hence it is relatively w^{∗}-sequentially compact (the weak star topology
being metrizable on bounded sets). Therefore, there exists a subsequence of (fτ_{n})
(possibly depending uponω) not relabeled and an elementx^{∗}_{ω}∈X^{∗} such that

(fτ_{n}(ω))w^{∗}-converges to x^{∗}_{ω}.

Definef_{∞}(ω) :=x^{∗}_{ω} forω∈Ω\N andf_{∞}(ω) := 0 forω ∈N. Then, taking into
account (5.6), we get

n→+∞lim hx`, f_{n}i=hx`, f_{∞}i=ϕ_{`} a.s. for all`≥1.

(5.7)

This implies the scalarF-measurability off_{∞}. Furthermore, one has
kf∞k ≤lim inf

n→+∞kfnk a.s.

which in view of (C_{2}) shows that kf_{∞}k is integrable. Thus f_{∞} ∈ L^{1}_{X}∗[X](F).

Next, replacingϕ_{`} in (5.5) withhx_{`}, f_{∞}i(because of the second equality of (5.7)),
we get

fn=E^{F}^{n}(f_{∞}) a.s. for all n≥1.

In particular, this yields sup

n

kfnk ≤sup

n

E^{F}^{n}kf∞k<∞ a.s.

(5.8)

Using the separability ofX, (5.7) and (5.8), we get
(f_{n}) w^{∗}-converges to f_{∞} a.s.

Finally, if the condition (T) is satisfied, then, reasoning as in the first proof (or the
first part of the second proof) of Theorem 5.1, we deduce that (f_{n})s^{∗}-converges

a.s. tof_{∞}.

We finish this section by extending Theorem 5.1 to mils. For this purpose the following decomposition result is needed [7, Corollary 3.1].

Proposition 5.3. Let (fn)n≥1 be a bounded mil in L^{1}_{X}∗[X](F). Then there
existsf∞∈L^{1}_{X}∗[X](F)such that

kfn−E^{F}^{n}(f_{∞})k →0 a.s. and
(fn) w^{∗}-converges to f_{∞} a.s.

Proof. As (f_{n}) is a bounded mil in L^{1}_{X}∗[X](F) for each x ∈X, (hx, f_{n}i) is a
bounded real mil inL^{1}

R(F), hence it converges a.s. to a functionrx∈L^{1}

R(F). On
the other hand, using [11, Theorem 6.1(4)], we provide an increasing sequence
(Ap)_{p≥1} in F with lim_{p→∞}P(Ap) = 1, a functionf_{∞}∈L^{1}_{X}∗[X](F) and a subse-
quence (f_{n}^{0})_{n≥1} such that

n→∞lim Z

Ap

hh, f_{n}^{0}idP =
Z

Ap

hh, f_{∞}idP

for all p≥1 and h∈ L^{∞}_{X}(F). By identifying the limit, we get rx =hx, f∞i a.s.

Thus

n→∞limhx, fn(ω)i=hx, f∞(ω)i a.s., (5.9)

for everyx∈X. So the real mil (hx, fn−E^{F}^{n}(f_{∞})i) converges to 0 a.s. Conse-
quently, it is possible to invoke an important result of Talagrand, ([34, Theorem 6])
which gives

kfn−E^{F}^{n}(f_{∞})k →0 a.s.

As

sup

n≥1

kE^{F}^{n}(f_{∞})k ≤sup

n≥1

E^{F}^{n}(kf_{∞}k)<∞ a.s.,
we deduce that

sup

n≥1

kfnk<∞ a.s.

Then, using (5.9), the separability ofX and the point-wise boundedness of (f_{n}),
we obtain the a.s. w^{∗}-convergence of (f_{n}) tof_{∞}.
Theorem 5.2. Let(fn)n≥1 be a bounded mil inL^{1}_{X}∗[X](F)satisfying the con-
dition (T). Then there existsf_{∞}∈L^{1}_{X}∗[X](F)such that

(fn) s^{∗}-converges a.s. to f∞.

Proof. The proof is similar to the one given in Theorem 5.1 by using Proposi-

tion 5.3 instead of Proposition 5.1.