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Abstract and Applied Analysis
Volume 2011, Article ID 582576, 7 pages
doi:10.1155/2011/582576
Research Article
On p-Convergence in Measure of a Sequence of
Measurable Functions
A. Boccuto,1 Ch. Papachristodoulos,2 and N. Papanastassiou2
1
Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1,
06123 Perugia, Italy
2
Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece
Correspondence should be addressed to N. Papanastassiou, npapanas@math.uoa.gr
Received 18 November 2010; Accepted 22 March 2011
Academic Editor: Agacik Zafer
Copyright q 2011 A. Boccuto et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
In the study by Papanastassiou and Papachristodoulos, 2009 the notion of p-convergence in
measure was introduced. In a natural way p-convergence in measure induces an equivalence
relation on the space M of all sequences of measurable functions converging in measure to zero.
We show that the quotient space M is a complete but not compact metric space.
1. Introduction
Convergence in measure plays a fundamental role in several branches of Mathematics, for
example in integration theory and in stochastic processes. In 1 a “Bochner-type” integration
theory was developed in the context of Riesz spaces with respect to a convergence introduced
axiomatically, and in particular some Vitali convergence theorems and Lebesgue dominated
convergence theorems were proved. Similar subjects were investigated by Haluška and
Hutnı́k 2, 3 in the setting of operator theory for Bochner- and Dobrakov-type integrals see
also 4, 5 and in 6–8 for the Kurzweil-Henstock integral in Riesz spaces.
In several contexts of integration theory it could be advisable to extend the concept of
convergence in measure in order to get applications, for example, in the study of the stochastic
integral and stochastic differential equations see, e.g., 9.
This paper is a continuation of 10, where the notion of p-convergence in measure
was introduced. In this paper we investigate a structure related to the vector space M of all
converging sequences of measurable functions.
Let Γ, Σ, μ be an arbitrary measure space, where μ is a 0, ∞-valued measure, and
let fn , f : Γ → , n 1, 2, . . ., be measurable functions.
2
Abstract and Applied Analysis
μ
We adopt the following usual terminology. By the notation fn → f we denote that the
sequence of measurable functions fn n converges in measure to f. Also for a pair fn n , f
and ε ≥ 0 we set
Aεn γ ∈ Γ : fn γ − f γ ≥ ε fn − f ≥ ε ,
n 1, 2, . . . .
1.1
We denote by the set of all positive integers and c0
the set of all real-valued nonnegative
sequences εn n converging to 0. Also for p > 0 we set
p
∞
p
εn < ∞ .
εn n : εn ≥ 0 for n 1, 2, . . . ,
1.2
n1
Convergence in measure is characterized by elements εn n of c0
as follows:
μ
fn −→ f iff there exists εn n ∈ c0
such that lim μ Aεnn 0.
1.3
n→∞
Taking into account that the sequence εn n above expresses the quality of approximation of
fn n to f, in 10 the authors introduced the following notion of convergence which we call
p-convergence in measure.
More precisely we say that, given p > 0, fn n p-converges in measure to f and we
p−μ
write fn −−−→ f if and only if there exists an element εn n ∈ p
such that
lim μ Aεnn 0.
1.4
n→∞
Obviously p-convergence in measure implies convergence in measure. It is proved see 10,
Preposition 2.3 that if the measure μ is not trivial and 0 < p < q, then p-convergence in
measure implies q-convergence in measure, while the converse implication in general fails. So
p-convergence in measure is strictly stronger than convergence in measure. As a consequence
of the above result we have that
M0
Mp Mq M∞ M,
0 < p < q,
1.5
where
M
Mp μ
fn n : fn −→ 0 ,
p−μ
fn n : fn −−−→ 0 ,
M0 p>0
Mp ,
M∞ p > 0,
1.6
Mp .
p>0
We note that M is considered as a vector space under usual operations and the notation
N M means that N is a proper vector space of M.
Abstract and Applied Analysis
3
2. Metric Spaces of Sequences of Measurable Functions
In a natural way p-convergence in measure induces an equivalence relation on the vector
μ
space M {f : f → 0}. We consider M as a subspace of L0 Γ , ℵ copies of the vector
n n
n
0
space L0 Γ of all real-valued measurable functions with the usual operations.
Definition 2.1. Let fn n , gn n be elements of M. We say that fn n , gn n are equivalent
fn n ∼ gn n if and only if for each positive real number p there exists an element εn n
of p
such that
lim μ fn − gn > εn 0
n→∞
2.1
or equivalently
p−μ
fn − gn −−−→ 0,
∀p > 0 ⇐⇒ fn − gn n ∈ M0 .
2.2
Since M0 is a vector subspace of M the relation ∼ is an equivalence one. We set M M ∼ M/M0 .
In the sequel we will define a metric d on M under which M turns to be a complete
metric space, similarly as a Fréchet space.
Definition 2.2. Let fn n ∈ M. We define
fn arctan inf A fn ,
n
2.3
p−μ
A fn p > 0 : fn −−−→ 0 p > 0 : fn n ∈ Mp .
2.4
where
By 1.5 it follows that Afn is an interval in .
Remarks 2.3. i We note that the above set Afn could be empty. In this case we set fn n π/2.
ii If fn n ∼ gn n , then fn n gn n .
Indeed, as gn n gn − fn n fn n and fn n ∈ Mp , it follows that fn n ∈ Mp if and only
if gn n ∈ Mp .
iii We set Mp Mp / ∼, p > 0, and hence Mp is a proper vector subspace of M and
the following strict inclusion holds:
Mp1
Mp
2
Proposition 2.4. The function : M →
i fn n ≥ 0
ii fn n 0 iff fn n ∼ 0n
if 0 < p1 < p2 see 1.5.
satisfies the following properties:
2.5
4
Abstract and Applied Analysis
0
iii afn n fn n for a /
iv fn gn n ≤ fn n gn n .
Hence, M becomes a metric space and the metric dfn n , gn n fn − gn n is
invariant under translations.
Proof. i It is obvious.
ii If fn n ∼ 0n , then fn n 0.
p−μ
Conversely, if fn n 0, then, fn n −−−→ 0 for each p > 0, and hence fn n ∈ M0 and
consequently fn ∼ 0n .
iii For a /
0, it holds that
Aεnn fn ≥ εn afn ≥ |a|εn ,
∞
p
εn < ∞ ⇐⇒
n1
∞
|a|εn p < ∞,
2.6
n1
for each sequence εn n of positive real numbers. Hence,
p−μ
fn n −−−→ 0
p−μ
iff afn n −−−→ 0,
2.7
which means that fn n afn n .
iv The inequality is obvious if fn n π/2 or gn n π/2.
Suppose fn n ≤ gn n < π/2. Then we conclude that Agn ⊂ Afn . Hence,
p−μ
p−μ
gn n −−−→ 0 implies fn n gn n −−−→ 0. So Agn ⊆ Afn gn , which implies that
fn gn ≤ gn ≤ fn gn .
n
n
n
n
2.8
Theorem 2.5. The space M, d is a complete metric space.
Proof. Let Fn n be a Cauchy sequence in M, where Fn fn,i i , n 1, 2, . . .. Hence, there
exists an increasing sequence of positive integers nk k such that
1
Fn − Fm < arctan ,
k
for n, m ≥ nk , k 1, 2, . . . .
2.9
This means that, for each n, m ≥ nk , there exists a sequence εn,m,i i of positive real numbers
1/k
with ∞
i1 εn,m,i < ∞ such that
μAn,m,i −→ 0,
i −→ ∞, where An,m,i fn,i − fm,i ≥ εn,m,i .
2.10
Then,
∞ i1
1/
max εn,nk
1 ,i
n ≤ n≤ nk
1
< ∞,
for 1, 2, . . . , k, k ∈ .
2.11
Abstract and Applied Analysis
5
From 2.10 and 2.11, proceeding by induction, it follows that there exists an increasing
sequence ik k of positive integers such that for each k we have
μAn,nk
1 ,i <
∞ iik
1
k
for i ≥ ik , n1 ≤ n ≤ nk
1 , k 1, 2, . . . ,
1/
max εn,m,i
<
n ≤ n≤ nk
1
1
,
2k
1
1
μ fnk
1 ,i ≥
< ,
k
k
2.12
for 1, 2, . . . , k,
2.13
for i ≥ ik ,
2.14
which express the uniform convergence to zero of finite number of sequence which converges
to zero and a finite number of tails of convergence series.
We set
F fn1 ,1 , fn1 ,2 , . . . , fn1 ,i1 −1 ; fn2 ,i1 , . . . , fn2 ,i2 −1 ; . . . ; fnk
1 ,ik , . . . , fnk
1 ,ik
1 −1 ; . . .
fi i .
2.15
By 2.14 it follows that F fi i ∈ M.
We have to show that
Fn − F −→ 0,
1/−μ
n −→ ∞ ⇐⇒ ∀ ∈ ∃n0 ∈ : fn,i − fi i −→ 0, for n ≥ n0 .
This means that we have to find n0 ∈
1/
< ∞ such that
εi i with ∞
i1 εi
2.16
and for n ≥ n0 a sequence of positive real numbers
μAi −→ 0, i −→ ∞, where Ai fn,i − fi ≥ εi .
2.17
Indeed let ∈ , n ≥ n0 n , and n ≤ nk < n < nk
1 for some k ∈ .
We set
εi 1,
if i 1, 2, . . . , ik − 1,
εi εn,nk
1 ,i ,
εi εn,nk
2 ,i ,
if i ik , . . . , ik
1 − 1,
2.18
if i ik
1 , . . . , ik
2 − 1,
and so on.
It holds that
∞
1
1
εi1/ ≤ ik − 1 k k
1 · · · < ∞
2
2
i1
by2.13 .
2.19
Also, for im ≤ i ≤ im
1 , m ≥ k, we have that
Ai fn,i − fi ≥ εi fn,i − fnm
1 ,i ≥ εn,nm
1 ,i
2.20
6
Abstract and Applied Analysis
by definition of fi . Hence, by 2.12, we take
μAi <
1
.
m
2.21
This implies 2.17, and the proof is complete.
Remarks 2.6. It is easy to see the following
a The addition : fn n , gn n → fn gn n is continuous.
b The translation Tgn n : fn n → fn n gn n fn gn n is a homeomorphism.
Hence, the system of neighborhoods of 0n determines the topology of M, d.
c The multiplication operator
Ha : fn n −→ a · fn n ,
a/
0,
2.22
is a homeomorphism.
d The multiplication a, fn n → afn n is not continuous. If an → 0, an /
0, n 0, and Fn F for n 1, 2, . . ., it holds that
1, 2, . . ., and F fn n ∈ M, F /
an −→ a / 0,
d
Fn −→ F,
2.23
but an Fn − 0F an Fn Fn F 0.
e The family Mp p>0 is a system of neighborhoods of 0n . Indeed, if Sr S0n , r {fn n : fn n < r} for r > 0, then for 0 < r1 < p < r2 we have
Sr1 ⊂ Mp ⊂ Sr2 .
Though M, d is not a topological vector space, M, d is complete and the subspaces
Mp , p > 0 constitute a system of closed and convex neighborhoods of 0n , as we will see in
the sequel Proposition 2.7. Hence, M, d is something like a Fréchet space. For example,
the principle of uniform boundedness holds true, as for this principle only continuity of Ha
is needed see 11.
Proposition 2.7. The subspaces Mp are closed for each p > 0.
Proof. Suppose that p > 0 and Fn n is a sequence in Mp , where Fn fn,i i , n 1, 2, . . ., and
F fn n ∈ M such that
d
Fn −→ F ⇐⇒ Fn − F −→ 0,
n → ∞.
2.24
Hence, there exist p < p and n0 such that
p −μ
Fn0 − F −−−−→ 0.
This implies that Fn0 − F ∈ Mp ⊂ Mp , and, since Fn0 ∈ Mp , it follows that F ∈ Mp .
2.25
Abstract and Applied Analysis
Proposition 2.8. M∞ p>0
7
Mp is a closed subspace of M.
Proof. Suppose F0 f0,i i ∈
/ M∞ , then F0 Mq , q > 0 is a neighborhood of F0 and F0 Mq ∩
Mp ∅ for all p > 0.
Indeed, if F0 F1 F2 for some F1 ∈ Mq and some F2 ∈ Mp , then F0 ∈ Mr , where
r maxp, q, which is a contradiction. Hence, F0 Mq ∩ M∞ ∅, which implies that M∞
is closed.
Remark 2.9. If S0n , r denotes the open sphere with center 0n and radius r, it is easy to see
that the family
{S0n , r}r>0 ∪ {F S0n , r}F ∈/M∞ , r > 0
2.26
is an open covering of M without a finite subcovering. Hence M is not compact.
Acknowledgment
N. Papanastassiou work is supported by the Universities of Perugia and Athens.
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