Kondurar Theorem in Riesz spaces
A. Boccuto
Dipartimento di Matematica
e Informatica
Università degli Studi
di Perugia
[email protected]
Abstract
In this paper we give a version of the
Kondurar theorem for functions taking values in Riesz spaces with respect to a general notion of convergence. If the Riesz space involved
is a space of random variables, the
most common types of convergence
are included. Some comments and
possible applications conclude the
paper.
Key words: Riesz spaces, convergence,
Riemann-Stieltjes integration, Kondurar theorem, Itô formula.
1
Introduction
In this paper we investigate the Kondurar theorem in Riesz spaces. The classical (scalar)
version of the Kondurar theorem asserts that,
whenever f and g are two real-valued functions, defined on a compact interval
[a, b] ⊂
R
IR, the Riemann-Stieltjes integral ab f dg exists, as soon as f and g are Hölder-continuous,
with orders α and β respectively, with α+β >
1.
Here we have chosen to adopt the harmless restriction that [a, b] = [0, 1], and to fix
the Riesz space setting as a product triple
(R1 , R2 , R) such that f is R1 -valued, g is R2 valued, and the integral takes values in R.
We formulated Riemann-Stieltjes integration
with respect to an interval function q, because
this is a more general approach, and in our
opinion is more suitable for applications.
D. Candeloro
Dipartimento di Matematica
e Informatica
Università degli Studi
di Perugia
[email protected]
Beyond the differences deriving from the
Riesz space setting, and from the general
notion of convergence here introduced ((L)convergence), our approach differs from the
classical one (see [6]) because we chose to split
the main proof into a number of partial results, whose purpose is to reduce the proof to
considering very special decompositions (decompositions with rational points, decompositions with intervals of the same length, dyadic
decompositions).
We think that this approach better clarifies
the importance of continuity properties, and
explains the precise moment in which the crucial property α + β > 1 plays its role (mainly,
in dealing with dyadic decompositions).
Moreover, the choice of a general interval
function q makes the results more interesting
even in the real-valued case.
In a previous paper (see [4]), a different (less
general) version of Kondurar’s theorem has
been found and applied in order to obtain
some kinds of Itô formula; other consequences
will be deduced in a forthcoming paper, so
we don’t present applications here. We just
observe that the classical stochastic integrals
(Itô, Stratonovich, Backward) and the classical Itô formula are included.
In Section 2 we introduce some preliminary
definitions and notations and prove some results, needed for the subsequent investigations. In Section 3 we present our version of
the Kondurar theorem, concluding with some
comments.
2
Preliminaries
In this section we introduce the definitions
used in proving the Kondurar theorem in
Riesz spaces. As we noticed in the Introduction, we shall consider only functions defined
in the interval [0, 1] . A Riesz space R is said
to be Dedekind complete if every nonempty
subset A ⊂ R, bounded from above, has
supremum in R. A unit of a Riesz space R
is an element u ∈ R such that u ≥ 0, u 6= 0.
From now on, we always suppose that R is a
Dedekind complete Riesz space.
In such a Riesz space R, let T be the set of all
bounded sequences (rn )n in R. In the literature, there exist several types of convergence
in Riesz spaces (see [7], [10]). In this paper we
give an axiomatic approach to convergence.
Definition 2.1 A convergence is a pair
(S, L), where S is a subspace of T and L is
a map L : S → R, satisfying the following
axioms:
a) L((ζ1 rn + ζ2 rn )n ) = ζ1 L((rn )n ) +
ζ2 L((sn )n ) for every pair of sequences
(rn )n , (sn )n ∈ S and for each ζ1 , ζ2 ∈ IR.
b) If (rn )n satisfies rn = l definitely, then
(rn )n ∈ S and L((rn )n ) = l.
c) Given three sequences, (rn )n , (sn )n , (tn )n ,
satisfying (rn )n , (tn )n ∈ S, L((rn )n ) =
L((tn )n ) = r, if rn ≤ sn ≤ tn definitely,
then (sn )n ∈ S and L((sn )n ) = r.
d) If u ∈ R, u ≥ 0, then the sequence
belongs to S and L
1
nu n
1
nu n
= 0.
By means of this concept, we define also convergence for nets, as follows.
Given any bounded net Ψ : ÷ → R, we say
that it is L-convergent to the limit λ, if there
exists a positive sequence (rn )n in R, belonging to S and satisfying L((rn )n ) = 0, such
that for every n it is possible to find an element ξ ∈ ÷, ξ = ξ(n), for which one has
|Ψ(ξ 0 ) − λ| ≤ rn for every ξ 0 > ξ.
It is easy to check that (O)-, (r)- and (∗)convergence satisfy axioms a),. . . ,d) in 2.1
(see also [7], [11]). Moreover, if (X, B, P ) is
a probability space, and R = L0 (X, B, P ) is
the Riesz space of all measurable functions,
with the identification up to sets of probability zero, then (O)- and (r)-convergence coincide with almost everywhere convergence, and
(∗)-convergence is equivalent to convergence
in probability (see [10]). We now introduce
some structural assumptions, which will be
needed in the construction of the RiemannStieltjes integral.
Assumptions 2.2 Let R1 , R2 , R be three
Dedekind complete Riesz spaces. We say that
(R1 , R2 , R) is a product triple if there exists a
map · : R1 × R2 → R, which we will call product, such that
2.2.1) (r1 + s1 ) · r2 = r1 · r2 + s1 · r2 ,
2.2.2) r1 · (r2 + s2 ) = r1 · r2 + r1 · s2 ,
2.2.3) [r1 ≥ s1 , r2 ≥ 0] ⇒ [r1 · r2 ≥ s1 · r2 ],
2.2.4) [r1 ≥ 0, r2 ≥ s2 ] ⇒ [r1 · r2 ≥ r1 · s2 ]
for all rj , sj ∈ Rj , j = 1, 2;
2.2.5) if Λ is any nonempty directed set,
(aλ )λ∈Λ is any net in R2 and b ∈ R1 , then
[aλ ↓ 0, b ≥ 0] ⇒ [b · aλ ↓ 0];
2.2.6) if (aλ )λ∈Λ is any net in R1 and b ∈ R2 ,
then [aλ ↓ 0, b ≥ 0] ⇒ [aλ · b ↓ 0].
We now give some examples of product
triples:
1)R1 = R is a Dedekind complete Riesz space,
R2 = IR, and the product involved is the
”scalar product”;
2)R1 = R2 = R = C(Ω), where C(Ω) is the
space of all real-valued continuous function
defined on Ω, and Ω is a compact topological space, such that the closure of any open
set is still an open set;
3)R1 = X, R2 = L(X, Y ), R3 = Y , where
X and Y are two Dedekind complete Riesz
spaces, and L(X, Y ) is the space of all (order)
continuous linear operators, defined on X and
taking values in Y . (Put x · f ≡ f (x), ∀ x ∈
X, ∀ f ∈ R2 ).
Now, let us denote by {I} the family of all
(nontrivial) closed subintervals of the interval [0, 1], and by D the family of all finite
decompositions of [0, 1] into pairwise nonoverlapping elements from {I}. Usually, any
such decomposition is denoted by a symbol
like D, and we shall write D = {0 = t0 < t1 <
. . . < tn−1 < tn = 1} or also D = {I1 , . . . , In },
where Ii = [ti−1 , ti ] for each i = 1, . . . , n.
Given any decomposition D = {I1 , . . . , In },
the mesh of D is the number δ(D) :=
max{|Ii | : i = 1, . . . , n}, where |I| as usual
denotes the length of the interval I. Clearly
the mesh defines a natural filtering order relation among all decompositions: we shall say
that a decomposition D1 precedes a decomposition D2 if δ(D2 ) ≤ δ(D1 ).
Another different ordering will be considered:
for any two decompositions, D1 , D2 , we say
that D2 is finer than D1 if every interval
from D2 is entirely contained in some interval from D1 . We shall denote this fact by
writing D2 ref D1 (D2 refines D1 ). Moreover,
we shall say that a decomposition D is rational if all its endpoints are rational, and that a
decomposition D is equidistributed if all intervals in D have the same length (clearly, if D
is equidistributed, then it is also rational). In
case D is equidistributed, and consists of 2n
intervals, we shall say that D is dyadic of order n. Of course, equidistributed and dyadic
decompositions of any interval [a, b] are similarly defined. In general, the family of all
decompositions of an interval [a, b] will be denoted by D[a,b] .
We now introduce the concept of integral of an
interval function q, according with the definition given in [4]. We shall skip the proof of
the main properties of such integral, because
they can be found in [4].
Definition 2.3 Let L be a convergence as in
2.1. Assume that q : {I} → R is any interval function. We say that q is (L)-integrable,
if there exists an element Y ∈ R such that
P
L((sup{|Y − I∈D q(I)|: D ∈ D, δ(D) ≤
1
n })n ) = 0. The element Y is called the (L)integral of q, and, when
no confusion can arise,
R
we will write Y := q.
Recalling our definition of L-convergence for
nets, the integral Y is nothing but the LP
limit of the net S(D) = I∈D q(I) where the
decompositions are ordered according to the
mesh : D1 > D2 iff δ(D1 ) ≤ δ(D2 ).
Definition 2.4 We say that an interval function q is Hölder-continuous if there exist a
unit u ∈ R and a positive real constant γ,
such that |q(I)| ≤ |I|γ u for all subintervals
I ⊂ [0, 1]. If this is the case, we say also that
q is Hölder-continuous of order γ.
A very useful tool to prove (L)-integrability is
the Cauchy property, according with the next
theorem: a more complete formulation (and a
proof) is given in [4] (see also [5]).
Theorem 2.5 Assume that q : {I} → R is
any interval function. The following conditions are equivalent:
(i) q is (L)-integrable;
(ii) the sequence (rn )n := (sup{|S(D) −
S(D0 )|: D, D0 ∈ D, δ(D0 ) ≤ n1 ,
Dref D0 })n is (L)-convergent to 0.
Other important facts are the following: if an
interval function is integrable in [a, b], then it
is integrable in any subinterval, and the integral is an additive function. These results are
summarized in the following theorem.
Theorem 2.6 Let q : {I} → R be an integrable function. Then, for every subinterval J ⊂ [0, 1], the function qJ is integrable,
where qJ is defined as qJ (I) = q(I ∩ J),
as soon as I ∩ J is nondegenerate, and 0
otherwise. Moreover, if {J1 , J2 } is any decomposition
of some
interval J ⊂ [0, 1],R we
R
R
R
have J q = J1 q + J2 q (Here, as usual, J q
means the (L)-integral
of qJ ). Finally, denotR
ing Z(I) := q(I) − I q for every interval I,
it turns out that |Z| is integrable, and it has
null integral.
Let now (R1 , R2 , R) be a product triple of
Riesz spaces. Usually, given a bounded function q : {I} → R2 , for every subinterval [a, b] ⊂ [0, 1] we set: ω(q)([a, b]) :=
sup{|q([u, v])| : a ≤ u < v ≤ b}. The
interval function ω(q) is called the oscillation of q. Moreover, given a bounded function g : [0, 1] → R2 , we can associate with g
a bounded interval function ∆(g), as follows:
∆(g)([a, b]) := g(b) − g(a), and so we can
also define ω(g)([a, b]) := ω(∆(g))([a, b]) =
sup{|∆(g)([u, v])| : a ≤ u < v ≤ b}
(∆(g)([a, b]) is also called the jump of g in
[a, b], while ω(g)([a, b]) is called the oscillation
of g in [a, b]). As usual, g is said to be Höldercontinuous of order α (with 0 < α ∈ IR) if
the function ∆(g) is.
Now we introduce the concept of RiemannStieltjes integral, with respect to an interval
function q. Again, details and proofs can be
found in [4].
Definition 2.7 Assume that f : [0, 1] →
R1 and q : {I} → R2 are two functions.
We say that f is (L)-Riemann-Stieltjes integrable with respect to q, if there exists
an element Y ∈ R, such that L((sup{|Y −
P
I∈D f (τI )q(I)|: τI ∈ I, D ∈ D, δ(D) ≤
1
n })n ) = 0. In case f is (L)-Riemann-Stieltjes
integrable w.r.t. q, the element Y ∈ R is
called the (L)-Riemann-Stieltjes
integral and
R1
denoted by Y := (L)(RS) 0 f dq. When we
have q = ∆(g), for some suitable function
g : [0, 1] → R2 , integrability of f with respect
to q will be expressed by saying that f is (L)Riemann-Stieltjes integrable w.r.t. g, and the
corresponding (L)-Riemann-Stieltjes Rintegral
will be denoted by Y := (L)(RS) 01 f dg.
When in particular R2 = IR and g(x) = x,
integrability of f with respect to g is called
Riemann-integrability.
The integral Y is the L-limit of the net
P
S(D) = I∈D f (τi ) q(I) (uniformly with respect to the choice of τI ∈ I), where the
decompositions are ordered according to the
mesh.
We have easily the following:
Proposition 2.8 If f : [0, 1] → R is Höldercontinuous of order γ, with γ > 0, and L is
a convergence as in 2.1, then f is Riemannintegrable.
3
The Kondurar theorem in Riesz
spaces
We begin with the following condition.
Definition 3.1 Let f : [0, 1] → R1 and q :
{I} → R2 be two fixed functions. We say
that f and q satisfy assumption (C) if the
interval function φ : {I} → R, defined as
φ(I) = ω(f )(I)|q(I)|, is (L)-integrable, and
its integral is 0.
The following result is easy (see also [4]).
Proposition 3.2 Let f : [0, 1] → R1 and q :
{I} → R2 be two bounded functions, satisfying assumption (C). ThenR the (L)-RiemannStieltjes integral (L)(RS) 01 f dq exists in R
if and only if the interval function Q(I) =
f (aI )q(I) is (L)-integrable, where aI denotes
the left endpoint of I.
In the next theorems, we shall assume in
addition that the interval function q is (L)integrable, according with Definition 2.3.
These theorems differ from similar results obtained in [4], because there q is required to be
additive (of course, if q is additive, it is trivially integrable, but the converse in general is
not true).
According with the previous theorem
2.6,
R
the integral function ψ(I) = I q turns
out to be additive, and the interval function |Z| = |q − ψ| has null integral.
The next step will be proving that, provided q is integrable, f and q are Höldercontinuous and assumption (C) is satisfied,
existence of the
(L)-Riemann-Stieltjes inteR1
gral (L)(RS) 0 f dg can be checked considering only rational decompositions.
Proposition 3.3 Assume that f : [0, 1] →
R1 and q : {I} → R2 are two Höldercontinuous functions, for which assumption (C) is satisfied.
Suppose also that
q is (L)-integrable, and that L((ρn )n ) =
P
0, where ρn = sup{| J∈D f (aJ )q(J) P
I∈D0 f (aI )q(I)|: D0 , D ∈ D, D0 , D rational, δ(D0 ) ≤ n1 , Dref D0 } for every n ∈ IN .
Then, there exists the (L)(RS)-integral of f
w.r.t. q.
Proof.
Since assumption (C) is satisfied, from Proposition 3.2 it’s enough to
prove that the R-valued interval function
Q(I) := f (aI )q(I) (or, equivalently, Q(I) :=
f (aI )ψ(I)) is (L)-integrable.
For every integer
n, let us define
nP
o
1
σn := sup
|Z(I)|
:
D
∈
D,
δ(D)
≤
I∈D
n .
As we already observed, (σn )n is (L)convergent to 0. Moreover, by hypotheses,
there exist two units u1 ∈ R1 , u2 ∈ R2 and
two positive real numbers, α and β, such
that |f (b) − f (a)| ≤ u1 |b − a|α , |q([a, b])| ≤
u2 |b − a|β for all [a, b] ⊂ [0, 1]. It is easy to
deduce that |f | and |q| are bounded, so we
can denote by F and G two majorants for
them, respectively. As usual, for every decomposition D of [0, 1], we denote S(D) =
P
P
Q(I) = I∈D f (aI ) q(I), and S(D) =
PI∈D
P
I∈D Q(I) =
I∈D f (aI ) ψ(I). The main
step in the proof will be the following Claim.
Claim: For every n ∈ IN and every D ∈ D
with δ(D) < n1 , there exists a rational decomposition D0 such that δ(D0 ) < n1 and
|S(D) − S(D0 )| ≤
F u2 + Gu1
+ 2F σn . (1)
n
To this aim, fix n ∈ IN , and take a decom1
position D, with δ(D) < . Set D := {0 =
n
t0 , t1 , . . . , tN , 1 = tN +1 }, and let θ = n1 −δ(D).
Choose now any positive number ε such that
1
1
, εβ < 2nN
. Now, for each
ε < θ, εα < nN
i = 1, 2, . . . , N choose a rational point τi such
that ti−1 < τi ≤ ti and ti − τi < ε. If
we define D0 := {0 = τ0 , τ1 , . . . , τN , 1}, it
is clear that D0 is a rational decomposition,
with δ(D0 ) < n1 .
Now we compute |S(D) − S(D0 )|, as follows:
S(D) − S(D0 ) = (f (0) − f (τ1 ))ψ([τ1 , t1 ])+
(f (t1 ) − f (τ1 ))ψ([t1 , τ2 ])+
(f (t1 ) − f (τ2 ))ψ([τ2 , t2 ]) + . . . +
(f (tN −1 ) − f (τN −1 ))ψ([tN −1 , τN ])+
(f (tN −1 ) − f (τN ))ψ([τN , tN ])+
(f (tN ) − f (τN ))ψ([τN , 1]).
From this, replacing throughout ψ(J) with
Z(J) + q(J), where Z = ψ − q, we obtain
|f (t1 ) − f (τ2 )|{|Z([τ2 , t2 ])| + |q([τ2 , t2 ])|}+
|f (t2 ) − f (τ2 )|{|Z([t2 , τ3 ])| + |q([t2 , τ3 ])|}+
|f (t2 )−f (τ3 )|{|Z([τ3 , t3 ])|+|q([τ3 , t3 ])|}+. . . +
|f (tN ) − f (τN )|{|Z([tN , 1])| + |q([tN , 1])|} ≤
2F σn + 2F N u2 εβ + N Gu1 εα .
Now we deduce that
|S(D) − S(D0 )|
F u2 + Gu1
,
n
i.e. the assertion of the Claim. From this, we
get
≤ 2F σn +
|S(D) − S(D0 )| ≤ |S(D) − S(D)|+
|S(D) − S(D0 )| + |S(D0 ) − S(D0 )| ≤
F u2 + Gu1
.
2F σn +|S(D)−S(D0 )| ≤ 4F σn +
n
Now we shall prove that Q is (L)-integrable.
According to the hypotheses, one easily sees
that sup{|S(D1 ) − S(D2 )|: D1 and D2 rational, δ(D1 ) < n1 , δ(D2 ) < n1 } ≤ 2ρn for
every n; hence, using the Claim above, we
see also that sup{|S(D) − S(D0 )|: D ∈ D,
D0 ∈ D, δ(D) < n1 , δ(D0 ) < n1 } ≤ 2ρn +
1
8F σn + 2 F u2 +Gu
for every n. Now, the asn
sertion follows from Theorem 2.5.
Up to the end of this section, we shall always
assume that q is integrable, that f and q are
Hölder-continuous of order α and β respectively, with related units u1 and u2 respectively. Moreover, we shall set γ = α + β and
assume that γ > 1.
As above, we shall define Q(I) := f (aI )q(I),
Q(I) := f (aI )ψ(I), Z(I) := ψ(I) − q(I),
for all intervals I ⊂ [0, 1], where aI denotes
the left endpoint of I, and ψ is the integral function of q. Finally, we shall define
P
σ(J) := sup{ I∈D |Z(I)|: D ∈ DJ } for every
sub-interval J.
Lemma 3.4 Let [a, b] be any sub-interval of
[0, 1], and let us denote by c its midpoint.
Then
|S(D) − S(D0 )| ≤
|f (t1 ) − f (0)|{ |Z([τ1 , t1 ])| + |q([τ1 , t1 ])|}+
|f (t1 ) − f (τ1 )|{|Z([t1 , τ2 ])| + |q([t1 , τ2 ])|}+
|Q([a, b]) − (Q([a, c]) + Q([c, b]))| ≤
b−a
2
α
u1 |Z([c, b])| + u1 u2
b−a
2
γ
.
Proof. We have
Similarly, we obtain
|
|Q([a, b]) − (Q([a, c]) + Q([c, b])| =
X
Q(J) −
J∈D2
≤ 2 u1 u2
|f (c) − f (a)|ψ([c, b])| ≤
u1
|f (c) − f (a)|{|Z([c, b])| + |q([c, b])|} ≤
≤(
|
Proposition 3.5 Let [a, b] be any subinterval
of [0, 1] and assume that D is a dyadic decomposition of [a, b], of order n. Then there exists
a positive element v ∈ R, independent of a, b
and n, such that
X
X
≤ 2 u1 u2
|
X
Proof. In this proof, for brevity we shall
write σ in the place of σ([a, b]). Let us
consider the dyadic decompositions of [a, b]
of order 1, 2, . . . , n and denote them by
D1 , D2 , . . . , Dn = D respectively. Then we
have
X
Q(I)|
|Q([a, b]) −
I∈D
Q(I)|+
I∈D1
X
Q(J) −
i=2 J∈Di
X
Q(I)|.
I∈Di−1
Thanks to Lemma 3.4, the first summand in
the right-hand side is less than
b−a
2
α
u1 |Z([c, b])| + u1 u2
b−a
2
where c is the midpoint of [a, b].
|Z([c, b])| ≤ σ, then we can write
|Q([a, b]) −
X
γ
,
≤
b−a
2
u1 σ + u1 u2
b−a
2
|Q([a, b]) −
b−a
4
γ
+ u1
X
Q(J) −
b−a
4
α
σ.
Q(I)|
I∈Di−1
b−a
2i
γ
+ u1
X
b−a
2i
Q(I)| ≤ u1 u2 {(
I∈D
α
σ,
b−a γ
) +
2
b−a
b−a γ
) + . . . + 2n−1 ( n )γ }+
4
2
b−a α
)
+u1 σ{(
2
b−a α
b−a
) + . . . + ( n )α }.
+(
4
2
The first summation is bounded by the quantity
i
n X
1
1
γ
u1 u2 (b − a)
2
2γ−1
i=1
2(
∞
X
1
1
≤ u1 u2 (b − a)γ
γ−1
2
2
i=1
=
1
2(2γ−1
− 1)
i
(b − a)γ u1 u2 .
1
u u , and turn to
Let us set v1 := 2(2γ−1
−1) 1 2
the second summation:
Q(I)|
α
Q(I)|
hence
Since
I∈D1
X
I∈D1
J∈Di
≤ u1 u2 2i−1
|
Q(J) −
In the same fashion, for every index i from 3
to n, we have
≤ v ((b − a)γ + (b − a)α σ([a, b]).
n
X
+
{|Z([c1 , c])| + |Z([c2 , b])|},
Q(I)|
X
γ
α
J∈D2
I∈D
≤ |Q([a, b]) −
b−a
4
where c1 is the midpoint of [a, c] and c2 is the
midpoint of [c, b]. Again, we have |Z([c1 , c])|+
|Z([c2 , b])| ≤ σ, so
b−a α
) u1 |Z([c, b])|+
2
b − a α+β
u1 u2 (
)
.
2
|Q([a, b]) −
b−a
4
Q(I)|
I∈D1
|f (a)ψ([a, b]) − f (a)ψ([a, c]) − f (c)ψ([c, b])| =
= |f (a)ψ([c, b]) − f (c)ψ([c, b])| ≤
X
γ
.
u1 σ{(
b−a α
)
2
+
b−a
4
α
+ ... +
b−a
2n
α
}
1
≤ u1 σ α
(b − a)α .
2 −1
1
Now, if we put: v = sup{v1 , 2αu−1
}, the assertion follows immediately.
In the next proofs, we shall again use the
P
notation: S(D) :=
I∈D Q(I), S(D) :=
P
Q(I)
for
every
decomposition
D.
I∈D
Proposition 3.6 Let [a, b] be any subinterval of [0, 1], and assume that D is an equidistributed decomposition of [a, b], consisting of
N elements. Then
|Q([a, b]) −
X
Q(I)|
I∈D
v
h X
2i
i=0
Proof. Of course, if N = 2h for some integer h, the result follows from Proposition
3.5. So, let us assume that N is not a power
of 2, and write N in dyadic expansion: N =
e0 20 + e1 21 + . . . + eh 2h , where (e0 , e1 , . . . , eh )
is a suitable element of {0, 1}h+1 , and h =
[log2 N ]. Clearly then, eh = 1. Now let
us define: t0 := a, th+1 := b, and tj :=
P
a + ji=1 ei−1 2i−1 b−a
N for j = 1, 2, . . . , h.
Thus we have tj − tj−1 = 0 or 2j−1 b−a
N according as ej−1 = 0 or 1 for j = 1, 2, . . . , h,
and in any case th = b − 2h b−a
N . Let us denote
by D0 the decomposition of [a, b] whose intermediate points are t0 , t1 , . . . , th+1 (dropping
duplicates if there are any: however we shall
keep the same notation, because this will not
affect the result). We observe that, for every
I ∈ D0 , the elements from D that are contained in I form a dyadic decomposition of I.
So we have, from Proposition 3.5:
|S(D0 ) − S(D)| ≤
X
X
|Q(I) −
I∈D0
v(|I|γ + |I|α σ(I)) ≤
≤v
h X
i=0
ib
2
−a
N
γ
+
2(h+1)γ
+
Nγ
2(h+1)α
σ([a, b]) ≤
N α (2α − 1)
2α
σ([a, b]).
2α − 1
α
Thus, setting w := v(max{2γ , 2α2−1 ), and recalling that b − a ≤ 1, we get
|S(D0 ) − S(D)| ≤ w ((b − a)γ + σ([a, b])) .
Let us now evaluate |Q([a, b]) − S(D0 )|. We
have
Q([a, b]) = f (a)ψ([a, b]) = f (a)ψ([a, t1 ])+
f (a)ψ([t1 , t2 ]) + . . . + f (a)ψ([th , b]),
hence
|Q([a, b])−S(D0 )| ≤ |(f (a)−f (t1 ))ψ([t1 , t2 ])+
(f (a) − f (t2 ))ψ([t2 , t3 ]) + . . . +
(f (a) − f (th ))ψ([th , b])| ≤
h
X
|f (a) − f (tj )| |ψ([tj , tj+1 ])| ≤
j=1
h
X
|f (a)−f (tj )| (|q([tj , tj+1 ])|+|Z([tj , tj+1 ])|) ≤
j=1
h
X
u1 u2 (tj − a)α (tj+1 − tj )β +
j=1
2F
h
X
|Z([tj , tj+1 ])|,
j=1
where F is any majorant for f .
Recall that tj+1 − tj ≤ 2j b−a
N , and so tj − a ≤
b−a
j−1 ) ≤ 2j b−a for all
(1
+
2
+
4
+
.
.
.
+
2
N
N
j = 1, 2, . . . , h. Hence
|Q([a, b]) − S(D0 )| ≤
h
X
j=1
I∈D0
σ([a, b]) ≤ v(b−a)γ
≤ v(b − a)γ 2γ + v(b − a)α
Q(J)| ≤
J∈D, J⊂I
X
α
v(b − a)α
≤ r((b − a)γ + σ([a, b]))
for some unit r ∈ R, independent of D and of
[a, b].
b−a
N
jb
u1 u2 2
−a
N
α jb
2
= u1 u2 (b − a)γ
−a
N
β
+2F σ([a, b]) =
h
1 X
2jγ +
N γ j=1
2F σ([a, b]) ≤ u1 u2 (b − a)γ 2γ
2hγ
+
Nγ
2F σ([a, b]) ≤ u1 u2 2γ (b − a)γ + 2F σ([a, b]).
2γ ,
Thus, setting v2 := u1 u2
we get |Q([a, b])−
γ
S(D0 )| ≤ v2 (b − a) + 2F σ([a, b]). Finally,
setting r = w + v2 + 2F , we deduce
|Q([a, b]) − S(D)| ≤ |Q([a, b]) − S(D0 )|+
|S(D0 ) − S(D)| ≤ r((b − a)γ + σ([a, b]).
We are now ready for the main result.
Theorem 3.7 (The Kondurar Theorem) Let
f and q be as above, with γ = α + β > 1.
Then f is (L)(RS)-integrable w.r.t. q.
Proof. We first observe that the real valued
interval function W (I) := |I|γ is integrable,
and its integral is null. By virtue of axiom d)
in 2.1, this immediately implies that f and q
satisfy assumption (C). Let r be the unit in
R given by Proposition 3.6, and set bk := 2r
k ,
∀ k ∈ IN . Now fix k and choose any rational
decomposition D0 such that δ(D0 ) ≤ k1 and
P
1
γ
J∈D |J| ≤ k for every decomposition D,
finer than D0 .
Now, if D is any rational decomposition, finer
than D0 , then there exists an integer N such
that the equidistributed decomposition D∗ ,
consisting of N subintervals, is finer than D.
Of course, D∗ is also finer than D0 , so we
have, by Proposition 3.6:
|S(D∗ ) − S(D0 )| ≤ |S(D∗ ) − S(D∗ )|+
|S(D∗ ) − S(D0 )| + |S(D0 ) − S(D0 )| ≤
≤
X
r|I|γ + rσk + 2F σk ≤
and the theorem is proved, thanks to Proposition 3.3.
As we already mentioned, if q is additive, then
q is integrable, and so the conclusion of the
previous theorem holds. But additivity of q
implies that q is the jump function of some
map g : [0, 1] → R2 , hence the last theorem contains also the following (more classical) version of the Kondurar theorem.
Corollary 3.8 Let f : [0, 1] → R1 and g :
[0, 1] → R2 be two Hölder-continuous functions, of order α and β respectively, and assume that α + β > 1. Then f is (L)(RS)integrable with respect to g.
Remark 3.9 Assume that W is any Stochastic Process, with Hölder continuous trajectories, on a probability space (X, B, P ), and
g : [0, 1] → L0 (X, B, P ) is the function which
associates to each t ∈ [0, 1] the random variable Wt ∈ L0 . We can endow the Riesz space
L0 with the convergence in probability (which
satisfies all assumptions in Definition 2.1); if
we assume that W is homogeneous and with
independent increments, that Wt ∈ L4 for all
t, and E(Wt ) = 0 for all t, then we deduce
that E(Wt2 ) is a linear function of t. Moreover, if E(Wt4 ) ≤ Kt2 for some positive constant k, one can prove that the interval function q(I) = (∆(g)(I))2 is integrable (see also
[8]. So, integrability of any other process f
with respect to ∆(g)2 can be deduced by Theorem 3.7.
References
[1]
A. BOCCUTO, Abstract Integration in Riesz
Spaces, Tatra Mountains Math. Publ. 5
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[2]
A. BOCCUTO, Differential and Integral Calculus in Riesz Spaces, Tatra Mountains
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[3]
A. BOCCUTO, A Perron-type integral of
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(2001), 185-204.
[4]
A. BOCCUTO, D. CANDELORO, E.
KUBIŃSKA, Kondurar Theorem and Itô formula in Riesz spaces, preprint (2003).
I∈D0
1
r + (r + 2F )σk ,
k
and similarly
|S(D∗ ) − S(D)| ≤
1
r + (r + 2F )σk .
k
Therefore, by axiom d) of 2.1, we have:
L ((sup {|S(D) − S(D0 )| : D, D0 are rational,
δ(D0 ) ≤ 1/k, Dref D0 })k ) = 0,
[5]
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16 (1996), 563-585.
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V. KONDURAR, Sur l’intégrale de Stieltjes,
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W. A. J. LUXEMBURG & A. C. ZAANEN,
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[8]
Z. SCHUSS, Theory and Applications of
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[9]
R. L. STRATONOVICH, A new representation for stochastic integrals and equations,
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