Illinois Journal of Mathematics
Volume 45, Number 1, Spring 2001, Pages 279–289
S 0019-2082
A CHARACTERIZATION OF VARIATIONALLY MCSHANE
INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
L. DI PIAZZA AND K. MUSIAL
0. Introduction
In [5] D.H. Fremlin studies, in a σ-finite outer regular quasi-Radon measure
space, a method of integration of vector-valued functions which is an essential
generalization of the McShane process of integration [8]. The method involves
infinite McShane partitions by disjoint families of measurable sets of finite
measure. For this integral Henstock’s lemma no longer holds. We therefore
consider the variational McShane integral (see [7]), which in general integrates
a family of vector valued functions that is a proper subset of the family of
McShane integrable functions. For a Banach-space valued function defined on
a closed interval endowed with the Lebesgue measure, the variational McShane
integral has been investigated in [1] by W. Congxin and Y. Xiabo (who called
the integral the strong McShane integral). Congxin and Xiabo showed that a
Banach-space valued function is variationally McShane integrable if and only
if it is Bochner integrable. Their proof is based on the Frechet differentiability
of the Bochner integral, and it cannot be directly generalized to an arbitrary
measure space. In this paper we prove the surprising result (Theorem 1 below)
that, in case of an arbitrary (even finite) quasi-Radon measure space, the
class of variationally McShane integrable functions can be significantly larger.
Our proof is rather elementary, but it gives a complete characterization of
variationally McShane integrable functions. In particular, it also yields a
simple proof of the result of Congxin and Xiaobo [1] (Theorem 2). Using this
characterization, we generalize a result of V. Skvortsov and V. Solodov [10]
showing that the McShane integral and the variational McShane integral are
equivalent, when considered on a closed interval equipped with the Lebesgue
measure, if and only if the range space is of finite dimension (Theorem 3).
Received October 20, 1999; received in final form January 7, 2000.
2000 Mathematics Subject Classification. Primary 28B05; Secondary 46G10.
c
2001
University of Illinois
279
280
L. DI PIAZZA AND K. MUSIAL
1. The variational McShane integral
Throughout this paper, X is a Banach space and (Ω, T , Σ, µ) is a σ-finite
and outer regular quasi-Radon measure space (see [4]); i.e., we have:
(i) (Ω, Σ, µ) is a σ-finite, complete and outer regular measure space;
(ii) T is a topology such that T ⊂ Σ;
(iii) µ(E) = sup{µ(F ) : F ⊆ E , F is closed} for every E ∈ Σ;
(iv) µ is τ −additive, i.e., if G ⊆ T is non–empty and upwards directed by
inclusion, then µ(∪G∈G G) = sup{µ(G) : G ∈ G}.
If f : Ω → X is Pettis Rintegrable with respect to µ, then we denote by
νf (E) the value of Pettis– E f dµ. As is well known (see [9]), the variation
|νf | of νf is a σ-finite measure that is absolutely continuous with respect to
µ (in the sense that if µ(E) = 0 then also |νf |(E) = 0). If |νf | is finite, then
it is also continuous in the ε − δ sense.
Henceforth, all integrals that are not explicitly described as Bochner integrals are to be understood as Pettis integrals,
A generalized partial McShane partition in Ω is a sequence h(Ei , ωi )i such
that hEi i is a family of pairwise disjoint sets of positive finite measure and
ωi ∈ Ω for each i. If µ(Ω \ ∪i Ei ) = 0 we call h(Ei , ωi )i a generalized McShane
partition of Ω. A function ∆ : Ω → T such that ω ∈ ∆(ω) for each ω ∈ Ω is
called a gauge. Given a gauge ∆ we say that a generalized (partial) McShane
partition h(Ei , ωi )i in Ω is ∆-fine if Ei ⊂ ∆(ωi ) for each i.
Definition 1. A function f : Ω → X is said to be variationally McShane
integrable (in short, VM-integrable) if it is Pettis integrable and for each ε > 0
there exists a gauge ∆ : Ω → T such that
Z
X
f (ωi )µ(Ei ) −
(1)
Ei
i
f dµ
<ε
for each ∆-fine generalized McShane partition h(Ei , ωi )i of Ω. The number
νf (Ω) is called the value of the variational McShane integral.
Note that (1) can be also written as
Z
X
f dµ/µ(Ei )
µ(Ei ) f (ωi ) −
< ε.
i
Ei
We begin by stating a few basic properties of the variational McShane
integral. The proofs are easy, and we do not present them here.
Proposition 1. Let (Ω, T , Σ, µ) be a σ-finite and outer regular quasiRadon measure space, and let X be a Banach space.
(a) If f : Ω → X is VM-integrable, then f is also McShane integrable
(see [5] for the definition), and the two integrals coincide.
MCSHANE INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
281
(b) If f : Ω → X is VM-integrable and A ⊆ Ω is nonempty, then f |A is
VM-integrable with respect to µ|A.
(c) If f, Rg : Ω → X are
then f + g is also VM-integrable
R VM-integrable,
R
and (f + g) = f + g.
(d) If Y is a Banach space and T : X → Y is a bounded linear operator,
then for each VM-integrable function
R f :Ω→
R X, the function T f :
Ω → Y is also VM-integrable with T f = T f .
(e) If f = 0 µ-a.e., then f is VM-integrable with its integral equal to
zero.
The proof of the main result will be broken into several lemmas. We start
with a result stating that Bochner integrable functions are variationally McShane integrable. The proof is similar to that of Fremlin for McShane integrability [5], so we do not present it here.
Lemma 1. If f : Ω → X is Bochner integrable, then f is also variationally
McShane integrable.
Definition 2. We say that a non-negative measure ν in Ω is moderated
(see [6]) if there exists a sequence hWn i of open sets of finite ν-measure covering Ω.
Lemma 2. If f : Ω → X is strongly measurable, Pettis integrable, and if
|νf | is moderated, then f is variationally McShane integrable.
Proof. It is well known that a strongly measurable and Pettis integrable
function f can be
represented in the form f = g + h, where g is Bochner
P∞
integrable, h = n=1 xn χAn , and the equality is understood to hold µ-a.e.
Moreover, the sets An can be taken to be pairwise disjoint and of finite measure. Since, by Lemma 1, g is variationally McShane integrable and its variation is moderated
(because it is finite), we may assume for simplicity that
P∞
f = h = n=1 xn χAn . Moreover, in view of PropositionP1, we may assume
∞
that f has countably
S many values and that we have f = n=1 xn χAn everywhere, with Ω = n An . In particular, this means that the xn need not be all
distinct. In fact, if f equals zero on a set of infinite measure, then zero has to
occur infinitely often among the values xn . Since |νf | is moderated, we may
assume that each of the sets An , n ∈ N, can be embedded into an open set
Wn of finite |νf |-measure. (Otherwise we take a sequence of open sets making
|νf | moderated and then take all possible intersections of these sets with the
sets An , n ∈ N. We denote these new sets also by An . It may be necessary
to modify the sets An , n ∈ N, on a set of µ-measure zero.)
Let ε > 0 be fixed. Since the measure |νf | is absolutely continuous with
respect to µ on each Wn , for each n ∈ N there exists ηn > 0 such that if
µ(E) < ηn and E ⊆ Wn , then |νf |(E) < ε/2n+1 . We may assume here that
ηn < ε/2 .
282
L. DI PIAZZA AND K. MUSIAL
Since µ is outer regular, for each n ∈ N there exists an open set Gn ⊃ An
such that Gn ⊆ Wn and
ηn
µ(Gn \ An ) < n
.
2 (kxn k + 1)
Then we have also
ε
|νf |(Gn \ An ) < n+1 .
2
Define a gauge ∆ : Ω → T such that
∆(ω) ⊂ Gn
if ω ∈ An .
Let h(Ei , ωi ) : i ∈ Ni be a ∆-fine generalized McShane partition. Note that,
while for each i there exists exactly one ni such that ωi ∈ Ani , it is possible
that ni = nj for some i 6= j. Since the partition is ∆−fine, we have Ei ⊆ Gni .
Let
Ci = Ei ∩ Ani
and
Di = Ei \ Ani .
Since f (ω) = xni whenever ω ∈ Ani , we have
Z
Z
X
X
f (ωi )µ(Ei ) −
f dµ =
[f (ωi ) − f (ω)] dµ(ω)
Ei
Ei
i
i
Z
Z
X
=
[f (ωi ) − f (ω)] dµ(ω)
[f (ωi ) − f (ω)] dµ(ω) +
i
Ci
Di
Z
X
=
[f (ωi ) − f (ω)] dµ(ω)
Di
i
Z
X
X
≤
kf (ωi )kµ(Di ) +
f
(ω)
dµ(ω)
i
i
Di
X Z
=
kf (ωi )kµ(Di ) +
f (ω) dµ(ω)
Di
n ωi ∈An
i
!
Z
X
[
X X =
kxn kµ
Di +
f (ω) dµ(ω)
.
X X
n
Now, since
S
n ωi ∈An
ωi ∈An
ωi ∈An
Di
Di ⊆ Gn \ An , we have
[
ηn
µ(
Di ) < n
2 (kxn k + 1)
ωi ∈An
and so
X
n
kxn kµ
[
ωi ∈An
Di
!
<
X ηn
n
2n
<
ε
.
2
Similarly, if ωi ∈ An , then Di ⊆ Gn \ An and so
X
Z
X X X ε
ε
≤
f
(ω)
dµ(ω)
|νf |(Gn \ An ) <
= .
n+1
2
2
Di
n
n
n
ωi ∈An
MCSHANE INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
283
Consequently,
Z
X
f (ωi )µ(Ei ) −
Ei
i
f dµ
< ε,
which proves the variational McShane integrability of f .
We next show the following result.
Lemma 3. If f : Ω → X is variationally McShane integrable then f is
strongly measurable.
Proof. If µ is purely atomic, then the strong measurability is a direct consequence of the Pettis measurability theorem (see [3]). Thus we may assume
that µ is non-atomic. Suppose that f : Ω → X is variationally McShane
integrable, but it is not strongly measurable. According to Corollary 3E
of [5] the closed linear space H generated by νf (Σ) is separable. For each
positive ε denote the set {ω ∈ Ω : inf x∈H kf (ω) − xk > ε} by Ωε . Notice
that, since f is not strongly measurable, there exists ε0 with µ∗ (Ωε0 ) > 0.
(Otherwise we would have f (ω) ∈ H for almost all ω, and by the Pettis measurability theorem this would imply the strong measurability of f .) The space
(Ωε0 , Σ ∩ Ωε0 , T ∩ Ωε0 , µ|Ωε0 ) is still an outer regular quasi-Radon measure
space and f |Ωε0 is VM-integrable. For simplicity of notations we will assume
that Ω = Ωε0 and µ(Ω) = 1. Thus, for each E ∈ Σ of positive measure and
each ω ∈ Ω, we have
Z
f dµ/µ(E)
(2)
> ε0 .
f (ω) −
E
On the other hand, since f is variationally McShane integrable, there exists
a gauge ∆ : Ω → T such that
Z
X
f dµ/µ(Ei )
(3)
µ(Ei ) f (ωi ) −
< ε0
i
Ei
for each ∆-fine generalized McShane partition h(Ei , ωi )i of Ω. Since the formulae (2) and (3) cannot be valid simultaneously, we get a contradiction,
which proves that f is strongly measurable.
Lemma 4.
moderated.
If f : Ω → X is variationally McShane integrable, then |νf | is
Proof. Suppose that |νf | is not moderated and that µ is positive on nonempty
P∞ open sets. As in the proof of Lemma 2, we may assume that f =
, where the sets An are pairwise disjoint and of finite mean
n=1 xn χAS
sure, Ω = n An , and f is Pettis integrable, but not Bochner integrable.
Then there exists at least one set An0 such that if U ⊃ An0 is open then
284
L. DI PIAZZA AND K. MUSIAL
R
kf k dµ = ∞. Let ε > 0 be a fixed real number. For each n ∈ N let
U \An0
Gn ⊃ An be an open set such that
ε
µ(Gn \ An ) < n+1
.
2
(kxn k + 1)
Let now ∆ : Ω → T be any gauge such that
∆(ω) ⊂ Gn
if ω ∈ An ,
and let h(Ei , ωi )ii∈I be an arbitrary ∆-fine partial McShane partition of Ω
with a finite index set I. Note that for each i there exists exactly one ni such
that ωi ∈ Ani . Since the partition is ∆−fine, we obtain in this case Ei ⊆ Gni .
Let
Ci = Ei ∩ Ani
and
Di = Ei \ Ani .
If ω ∈ Ani , then f (ω) = xni , and so
Z
X
f (ωi )µ(Ei ) −
f
dµ
Ei
i∈I
Z
X Z
[f (ωi ) − f (ω)] dµ(ω) +
=
i∈I
Ci
Di
[f (ωi ) − f (ω)] dµ(ω)
Z
X
=
[f
(ω
)
−
f
(ω)]
dµ(ω)
i
Di
i∈I
X
Z
X
−
≥
f
(ω)
dµ(ω)
kf (ωi )kµ(Di )
Di
i∈I
i∈I
X Z
X X
−
=
f
(ω)
dµ(ω)
kf (ωi )kµ(Di )
i∈I
Di
n ωi ∈An
X
Z
X
=
f (ω) dµ(ω)
kxn kµ
−
Di
n
i∈I
Z
X
ε
− ,
f
(ω)
dµ(ω)
>
2
i∈I
[
Di
!
ωi ∈An
Di
S
where the last inequality follows from the relation ωi ∈An Di ⊆ Gn \ An .
Thus we obtain
Z
Z
X
X
ε
f (ωi )µ(Ei ) −
≥
−
(4)
f
dµ
f
(ω)
dµ(ω)
2
i∈N
Ei
i∈N
Di
for an arbitrary ∆-fine generalized McShane partition h(Ei , ωi )i∞
i=1 of Ω.
e
Take now an arbitrary gauge ∆. We will show that, for this gauge and
with the value of ε that was fixed at the beginning of the proof, condition
(1) of the variational integrability of f does not hold, for a suitably chosen
MCSHANE INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
285
e
partition. Without loss of generality, we may assume that ∆(ω)
⊆ ∆(ω) for
every ω ∈ Ω. We then have
[
e
An0 ⊆
∆(ζ)
.
ζ∈An0
S
e
Set Un0 = Gn0 ∩ ζ∈An ∆(ζ)
and let hζi i be a sequence of points from An0
0
satisfying the equality
!
[
e i) = 0 .
µ Un \
∆(ζ
0
i
e 1 ) ∩ Un and Wi = ∆(ζ
e i ) ∩ Un \ S Wj , whenever i ≥ 2.
Let W1 = ∆(ζ
0
0
j<i
Without loss of generality we may assume that µ(Wi ) > 0 for all i. Then
e
e
h(Wi , ζi )i is a ∆-fine
generalized partition of Un0 . Let h(Fp , ξp )i be any ∆fine generalized partition of Ω \ Un0 . Combining these partitions we obtain a
e
generalized ∆-fine
partition h(Vr , υr )i of Ω. By assumption, f is not Bochner
integrable on Un0 \ An0 , so there exists a sequence
h(Hj )i of pairwise disjoint
S
subsets of Un0 \ An0 such that Un0 \ An0 = j Hj and (see [3])
Z
X
(5)
f dµ = ∞ .
Hj
j
We now modify the partition h(Vr , υr )i as follows. We leave the pairs
(Fp , ξp ) unchanged, but replace each pair h(Wi , ζi )i by the family of pairs
{(Wi ∩ Hj , ζi ) : µ(Wi ∩ Hj ) > 0} ∪ {(Wi ∩ An0 , ζi ) : µ(Wi ∩ An0 ) > 0}.
e
Rearranging this family in a single sequence h(Ei , ωi )i, we obtain a ∆-fine
generalized partition of Ω. Note that, for each ζi ∈ An0 , we have Di ⊆
Un0 \ An0 and so µ(Di ) = 0 or Di ⊆ Hj for some j. Moreover, we have


[
µ (Un0 \ An0 ) \
Di  = 0.
ζi ∈An0
Applying (4) and (5) we obtain
Z
X
f (ωi )µ(Ei ) −
f
dµ
Ei
i∈N
≥
Z
ε
ε X
f dµ − ≥
f dµ − = ∞ .
Hj
2
2
Di
j
X Z
ζi ∈An0
It follows that f is not variationally McShane integrable. This completes the
proof.
Combining the above lemmas, we obtain the main result of this paper:
286
L. DI PIAZZA AND K. MUSIAL
Theorem 1. A function f : Ω → X is variationally McShane integrable
if and only if it is strongly measurable and Pettis integrable, and |νf | is a
moderated measure.
As a corollary we obtain a generalization of the result proved in [1] for the
case when Ω = [0, 1], with the Lebesgue measure.
Theorem 2. Let (Ω, T , Σ, µ) be a compact finite Radon measure space.
Then a function f : Ω → X is variationally McShane integrable if and only if
f is Bochner integrable.
Proof. The “only if” part follows from Theorem 1 and from the observation
that on a compact space, each moderated measure is finite.
2. Examples
It was shown in [10] that if [0, 1] is considered with the Lebesgue measure and the ordinary topology, then McShane integrability is equivalent to
variational McShane integrability if and only if the Banach space is finite
dimensional. Since for a σ-finite outer regular quasi-Radon space and a separable Banach space McShane integrability is equivalent to Pettis integrability
(see [5]), Theorem 2 gives the following generalization of the result in [10]:
Theorem 3. Let (Ω, T , Σ, µ) be a σ-finite quasi-Radon measure space,
and let X be a Banach space. If there is an uncountable compact set K ⊆ Ω
of positive measure such that µ is Radon on K, then McShane integrability
is equivalent to variational McShane integrability if and only if the Banach
space is finite dimensional.
We do not know whether such a characterization holds in the case of an
arbitrary non-atomic measure space. The topology of Ω certainly plays an
important part here. In view of several results of [6] it is quite possible that
the answer depends on some special axioms. As far as atomic measures are
concerned, the following example shows that for some particular measures
these two types of integration may coincide in case of an arbitrary separable
Banach space.
Example 1. Consider the interval [0, 1] endowed with the discrete topology and with a measure µ that is concentrated on the rationals of [0, 1] and defined on the family of all subsets of [0, 1]. It is clear that the measure is quasiRadon and, for any Banach space X, every strongly measurable X-valued
µ-Pettis integrable function is variationally McShane integrable. Fremlin [5]
showed that in the case of separable Banach spaces the McShane integral and
the Pettis integral coincide. It follows that in the case of the above example
the variational McShane integral coincides with the McShane integral for any
separable Banach space.
MCSHANE INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
287
Example 2. Consider the same measure space as above, but with the
natural topology of [0, 1]. This is also a quasi-Radon measure space. We will
show, however, that for some particular measures µ not all Pettis integrable
functions are variationally McShane integrable. By Lemma 4, it is enough to
construct a measure µ and a function f :[0,1]→ X such that |νf | is not moderated. In fact, we will construct a more complicated example of a function f
such that |νf | is infinite on each non-empty open interval. Thus, in particular,
the function f will not be VM-integrable on any interval.
To begin the construction, let {rn : n ∈ N} be a fixed enumeration of
all rationals from [0, 1], and let (Bn )∞
n=1 be a fixed enumeration of all open
subintervals of [0, 1] with rational endpoints, such that each of the P
intervals
∞
occurs infinitely often. Given an infinite dimensional space X, let n=1 xn
be an unconditionally convergent series that is not absolutely convergent (see
[2]). Our construction is by induction.
In the first step, let {rki : i ∈ N} be the subsequence of {rn : n ∈ N}
(ordered in the same way as the entire sequence) consisting of all rationals
from B1 . Let m1 be the first index such that
m1
X
kxi k ≥ 1 .
i=1
We set
µ({rk1 }) = 1/2 , . . . , µ({rkm1 }) = 1/2m1
and
f1 =
X
2i xi χ{rki } .
i≤m1
We denote the set {rk1 , . . . , rkm1 } by W1 and set V1 = ∅. We then define a
function ϕ1 : W1 → N by setting ϕ(rki ) = i.
We now describe the general inductive step. Assume that the construction
has been completed for B1 , . . . , Bp . Thus, we have sets V1 , . . . , Vp of rationals,
pairwise disjoint sets W1 , . . . , Wp of rationals, a sequence m1 ≤ . . . ≤ mp of
integers and, for i ≤ p, functions ϕi : W1 ∪ · · · ∪ Wi → N and fi : W1 ∪ · · · ∪
Wi → X with the following properties:
S
(a) for all i ≤ p, Vi = j<i Wj ∩ Bi and Wi ∩ Vi = ∅;
P
(b) for all i ≤ p, r∈Vi ∪Wi kxϕi (r) k ≥ i ;
(c) for all i ≤ p, ϕi (W1 ∪ . . . ∪ Wi ) = {1, . . . , mi } ;
(d) for all r ∈ W1 ∪ . . . Wp , µ({r}) = 2−ϕp (r) ;
(e) for allPi < p, ϕi+1 |(W1 ∪ . . . Wi ) = ϕi ;
(f) fp = r∈W1 ∪···∪Wp 2ϕp (r) xϕp (r) χ{r} .
Let {rqi : i ∈ N} be the subsequence of those rationals from Bp+1 that
were not involved in the construction at any previous step, and let Vp+1 =
288
S
L. DI PIAZZA AND K. MUSIAL
P
Wj ∩ Bp+1 . If r∈Vp+1 kxϕp (r) k ≥ p + 1, then we continue the construction with Bp+2 . Otherwise, let k be the first number such that
X X
xϕ(r) +
kxmp +i k ≥ p + 1,
j≤p
r∈Vp+1
i≤k
set
mp+1 = mp + k ,
Wp+1 = {rq1 , . . . , rqk }
and
ϕp+1 (rqi ) = mp + i , ∀ i ≤ k
and ϕp+1 (r) = ϕp (r) if r ∈ W1 ∪ . . . ∪ Wp .
Next, set for each i ≤ k
µ({rqi }) = 2−ϕp+1 (rqi )
and
fp+1 =
X
2ϕp+1 (rqi ) xϕp+1 (rqi ) χ{rqi } .
i≤k
Now define
ϕ(r) = ϕp (r) if r ∈ Wp
and f (t) =
∞
X
fn (t) for all t ∈ [0, 1] .
n=1
P∞
Since the series n=1 xϕ(rn ) is unconditionally convergent, the function f is
Pettis integrable (see [3]). If B is anP
arbitrary open subinterval of [0, 1] with
rational endpoints, then |νf (B)| = r∈B 2ϕ(r) kxϕ(r) kµ({r}) (see [3]). But
since there exists an increasing sequence k1 < k2 < . . . such that B = Bk1 =
Bk2 = . . ., we then have, for each i ∈ N,
X
|νf |(B) = |νf |(Bki ) ≥
xϕki (r) ≥ ki .
r∈Vki ∪Wki
Hence |νf |(B) = ∞, and consequently the measure |νf | is not moderated.
These two examples show that the class of variationally McShane integrable
functions depends not only on the measure space, but on the topology making
the measure quasi-Radon.
Example 3. We present here an example of a quasi-Radon measure space
and a Banach-space valued function that generates a vector measure with
moderated variation, but which itself is not Bochner integrable. This will
show that Theorem 1 in general cannot be reduced to Bochner integrability.
Let Ω = [0, ∞), let Σ be the family of all Lebesgue measurable sets and let
µ be given by
∞
X
λ[E ∩ (2n, 2n + 1)]
µ(E) =
,
2n+1
n=0
MCSHANE INTEGRABLE BANACH-SPACE VALUED FUNCTIONS
289
where λ is the P
Lebesgue measure. Let X be an infinite dimensional Banach
space, and let n xn be an X-valued unconditionally convergent series that
is not absolutely convergent. By a standard argument one can split the series
into countably many absolutely convergent subseries with pairwise disjoint
sets of terms, say
∞
X
[[
xnki , n = 1, 2, ... and {xn : n ∈ N} =
{xnki } .
n
i=1
i
We define functions fn on the interval [0, ∞) by
∞
X
fn (t) = 2n+1
2i+1 xnki χ(2n+1/2i+1 ,2n+1/2i ) (t)
i=1
P∞
and put f (t) = n=1 fn (t). The function f is not Bochner integrable, but it
has a moderated variation.
Acknowledgements. This paper was written while the second author
was visiting the University of Palermo. The research of the first author was
supported by the M.U.R.S.T. of Italy, while the research of the second author
was supported by IndAM of Italy.
References
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L. Di Piazza, Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
E-mail address: [email protected]
K. Musial, Wroclaw University, Institute of Mathematics, Pl. Grunwaldzki
2/4, 50-384 Wroclaw, Poland
E-mail address: [email protected]