On The Mcshane Integral For Riesz‐Spaces‐Valued Functions Defined
On Real Line
Yosephus D. Sumanto
1
, Muslim ansori
2
1
Mathematics Departement, Universitas Diponegoro
Jln.Prof.H. Soedarto,SH,Tembalang,Semarang. Email :sumanto_123yahoo.com
2
Mathematics Departement, Universitas Lampung
Jln. Soemantri Brodjonegoro No. 1 Bandar Lampung. Email :
ansomathyahoo.com
Abstract
This paper is a partial result of our researchs in the main topic ʺOn The McShane Integral for Riesz‐
Spaces ‐valued Functions Defined on the space
ʺ. We construct McShane integral for Riesz‐spaces‐ valued
functions defined on the space by a technique involving double sequences and prove some
basic properties among which the fact that our new integral is coincides with the McShane Integral for
Banach ‐spaces valued functions defined on space .
n
R R
R
Keywords : Riesz Space, McShane Integral
1. Introduction
The recent results of Integral theory were the Henstock‐Kurzweil integral
for Riesz‐space‐valued functions defined on bounded subintervals of the real
line and with respect to operator‐valued measures was investigated by
Riecan1989,1992 and Riecan and Brabelova1996, with respect to
D
‐ convergence
that is a kind of convergence in which the
ε
‐technique is replaced by
a technique involving double sequences , see Riecan and Neubrunn1997, with
respect to the order convergence, see Boccuto1998 and in Boccuto and Riecan2004
with respect to the order convergence but the Henstock‐Kurzweil integral
for Riesz‐space‐valued functions was defined on unbounded subintervals
of the real line. The
main goal of this paper is to generalize the above results by constructing
McShane integral for Riesz‐valued functions defined on Euclidean space
by a technique involving double sequences.
R
Dipresentasikan dalam SEMNAS Matematika dan Pendidikan Matematika 2007 dengan tema “Trend Penelitian Matematika dan Pendidikan Matematika di Era Global” yang
diselenggarakan oleh Jurdik Matematika FMIPA UNY Yogyakarta pada tanggal 24 Nopember 2007
2. Preliminary
Let be the set of all strictly positive integers, the set of the real numbers,
be the set of all strictly positive real numbers. Let
N R
+
R
1 1
2 2
... a
a b
a b
b
.
[ ]
, a b
is called interval or cell. .
A collection of intervals
∈ =
⎡ ⎤
⎣ ⎦
, , ,
, ,
0,1,2,...,
i i
i i
i i
a b a b
R a b i
n
[ ]
,
i i
a b
is called nonoverlapping if their interiors are dijoint. An additive
function on a set is a function defined on the family of all
subintervals of
such that
⊂ ⎡
⎤ ⎣
⎦ ,
a b R
F
⎡⎣ , a b
⎤⎦ ∪
= +
F B C
F B F C
for each pair of nonoverlapping intervals
⊂ ⎡ ⎤
⎣ ⎦
, ,
B C a b
. A nonnegative additive
function on a set
is called length on
l ⎡⎣ ,
a b ⎤⎦
⎡ ⎤
⎣ ⎦
, a b
, defined by
[ ]
, a b
b a = −
l
. Let
be a collection of pairs of
{
=
1 1
2 2
, ,
, ,...,
,
r r
P A x
A x A x
}
,
i i
A x
, where
are nonoverlapping
intervals,
1 2
, ,...,
r
A A A
=
= U
1 r
i i
A A
and
δ
⊂ ,
i i
i
A O x
x
, where
δ
,
i i
O x x
is open ball
with center
i
x
and radius
δ
i
x
,
= 1,2,..., i
r
. We say that is
P
δ
− fine
McShane Partition
on .
A
The real vector space with elements
is called an ordered vector space
if is partially ordered which satisfy :
L
1 2
, ,...
E E L
≤ ⇒
+ ≤ +
1 2
1 2
E E
E h
E h
for every and
for every in . If, in addition, is lattice with
respect to the partial ordering, then is called a Riesz space. For example,
with the familiar coordinate wise addition and scalar multiplication. and by
coordinatewise ordering, i.e., for
∈ h
L ≥ ⇒
≥ E
kE ≥ 0
k R
L L
n
R
= =
1 1
,..., ,
,...,
n n
x x
x y
y y
, we define,
≤ x
y
whenever
≤ =
, 1,...,
k k
x y k
n
, then is a Riesz space.
n
R
Definition 2.1 Zaanen,1996 : A Riesz space is said to be Dedekind complete if
every nonempty subset of , bounded from above, has supremum in
L
.
L L
SEMNAS Matematika dan Pend. Matematika 2007 928
Definition 2.2 Riecan, 1998 : A bounded double sequence
∈
, ,
i j i j
a L
is called regulator
or
D
‐sequence if, for each
∈ ↓
,
,
i j
i N a
, that is
and .
+
≥ ∀ ∈
, ,
1 i j
i j
a a
j N
∈
∧ =
,
i j j
a
Definition 2.3 Boccuto and Riecan, 2004 : Given a sequence
∈
n n
r L
. Sequence
n n
r
is said to be
D
‐convergence to an element
∈ r
L
if there exist a regulator
, ,
i j i j
a
, satisfying
the following condition : for every mapping
ρ
→ : N
N
, denoted by
there exists an integer such that
ρ
∈
N
N n
ρ
∞ =
− ≤ ∨
, 1
n i
i i
r r
a
for all . In this case, the
notation is denoted by
.
≥ n
n
= lim
n n
D r
r
Definition 2.4 Boccuto and Riecan, 2004 : A Riesz Space is said to be weakly
L
σ
−
distributive if for every
D
‐ sequence
, i j
a
, then
ρ ρ
∞ =
∈
⎛ ⎞
∧ ∨ =
⎜ ⎟
⎝ ⎠
,
1 i
i i
a
.
Throughout the paper, we shall always assume that is Dedekind
complete weakly
L
σ
−
distributive Riesz space.
Main Results
Definition 3.1 : A function
[ ]
⊂ →
: ,
f a b
R L
is said to be McShane integrable denoted
by
[ ]
∈ l
, , ,
f M a b L
, if there exists an element
∈ E
L
and
D
‐sequence such that for every
we can find a function
∈
, ,
i j i j
a L
ρ
∈
N
N
[ ]
δ
+
→ :
, a b
R
such that
Matematika 929
ρ
∞ =
=
− =
− ≤ ∨
∑ ∑
l l
, 1
1 r
k r
i i
i k
P f x
I E
f x I
E a
for every
δ
‐fine
M
partition
[ ]
{ }
{ }
= =
1 1
2 2
, ,
, ,
, ,...,
,
r r
P a b x
I x I x
I x
on
[ ]
, a b
. We
note that the McShane integral with respect to is well‐ defined, that is
there exists at most one element , satisfying Definition 3.1 and in this case we
have
l
E
[ ]
=
∫
, a b
M fdx
E
. The uniqueness is given in the following theorem.
Theorem 3.2 : If a function
[ ]
α
∈ ,
, , f
M a b L
, then its integral is unique.
Proof:
Let
[ ]
∈ l
, , ,
f M a b L
. If both and are McShane integral of function
, satisfying Definition 3.1, then there exists two
1
E
2
E f
D
‐sequence
, ,
i j i j
a
and
, ,
i j i j
b
in such that for every ,
we can find two positive function
L
ρ
∈
N
N
δ
1
and
δ
2
on
[ ]
, a b
, respectively, and for every
1
δ
‐fine M‐ partition
[ ]
{ }
=
1
, ,
P a b x
and
2
δ
‐fine M‐partition
[ ]
{ }
=
2
, ,
P a b x
on , we have
A
ρ
∞ =
− ≤ ∨
∑
l
1 1
, 1
i i
i
P f x
I E
a
and
ρ
∞ =
− ≤ ∨
∑
l
2 2
, 1
i i
i
P f x
I E
b
respectively. Let now
{ }
δ δ
δ
=
1 2
min ,
x x
x
, for every
∈ x
A
and take any
δ
−
fine M‐ partition
[ ]
{ }
= ,
, P
a b x
on , then
A
[ ]
{ }
= ,
, P
a b x
is both
1
δ
‐fine M‐ partition
and
2
δ
‐fine Perron partition on
[ ]
, a b
, and thus we have
ρ ρ
ρ ρ
ρ
∞ ∞
= =
∞ =
∞ =
≤ −
≤ + −
+ −
≤ ∨ + ∨
≤ ∨ +
≤ ∨
∑ ∑
l l
1 2
1 1
2 2
, ,
1 1
, ,
1 ,
1 i
i i
i i
i i
i i
i i
i i
i
E E
P f x
I E
P f x
I E
a b
a b
c
where
= +
∀ ∈
, ,
,
2 ,
i j i j
i j
c a
b i j
N
. By arbitrariness of
, we get
ρ
∈
N
N
SEMNAS Matematika dan Pend. Matematika 2007 930
ρ ρ
∞ =
∈
⎛ ⎞
≤ −
≤ ∧ ∨ =
⎜ ⎟
⎝ ⎠
1
2 ,
1 i
i i
E E
c
since is
, i j
c
D
‐sequence and thanks to weak
σ
−
distributivity of . Thus
, and so our M‐integral is well‐defined.
L =
1
E E
2
`
Now, we give some fundamental properties of
l , ,
M A L
.
Theorem 3.3 : If
∈ l
1 2
, ,
f f M A L
,
and
∈
1 2
, k k
R
, then
and
+ ∈
l
1 1 2 2
, , k f
k f M A L
[ ] [ ]
[ ]
+ =
+
∫ ∫
1 1 2 2
1 1
2 2
, ,
a b a b
a b
M k f
k f dx k M
f dx k M
f d
∫
,
x
.
Proof : similar to that M‐integral with values in real spaces.
Theorem 3.4 : If
[ ]
∈ l
, ,
, f g
M a b L,
and
≤ f x
g x
for every
∈ x
A
, then
[ ] [ ]
≤
∫ ∫
, ,
a b a b
M fdx
M gdx
.
Proof
: By hypotesis, there exists two
− D
sequences,
, ,
i j i j
a
and
, ,
i j i j
b
such that,
for every ,
we can find positive functions
ρ
∈
N
N
1
δ
dan
2
δ
, respectively on ,
and whenever
A
[ ]
{ }
=
1
, ,
P a b x
is
δ
1
‐fine M‐partition and
[ ]
{ }
=
2
, ,
P a b x
is
δ
2
‐ fine
M‐partition on
[ ]
, a b
, we have
[ ] [ ]
[ ]
ρ ρ
ρ
∞ =
∞ ∞
= =
− ≤ ∨
⇔ − ∨
≤ ≤
+ ∨
∑ ∫
∑ ∫
∫
l l
1 ,
1 ,
1 ,
, 1
1 ,
, i
i i
a b i
i i
i i
i a b
a b
P f x
I M
fdx a
M fdx
a P
f x I
M fdx
a
and
[ ] [ ]
[ ]
ρ ρ
ρ
α α
∞ =
∞ ∞
= =
− ≤ ∨
⇔ − ∨
≤ ≤
+ ∨
∑ ∫
∑ ∫
∫
l l
2 ,
1 ,
2 ,
, 1
1 ,
, i
i i
a b i
i i
i i
i a b
a b
P f x
I M
gd b
M gdx
b P
g x I
M gd
b
Matematika 931
respectively. For
every
[ ]
∈ , x
a b
, let
{ }
1 2
min ,
x x
x
δ δ
δ
=
, and take
δ
‐fine M‐partition
[ ]
{ }
= ,
, P
a b x
on
[ ]
, a b
, then
[ ]
{ }
= ,
, P
a b x
is both
δ
i
‐fine M‐partition
= 1,2 i
on
[ ]
, a b
. Thus we get
[ ] [ ]
ρ ρ
∞ ∞
= =
− ∨ ≤
≤ ≤
+ ∨
∑ ∑
∫ ∫
l l
, ,
1 1
, ,
i i
i i
i i
a b a b
M fdx
a P
f x I
P g x
I M
gdx b
and hence, for every
,
ρ
∈
N
N
[ ] [ ]
ρ ρ
∞ ∞
∞ =
= =
− ≤ ∨
+ ∨ ≤ ∨
∫ ∫
, ,
1 1
1 ,
, i
i i
i i
i i
i i
a b a b
M fdx
M gdx
a b
c
ρ
,
where
= +
∀ ∈
, ,
,
2 ,
i j i j
i j
c a
b i j
N
. By arbitrariness of
, since is a
ρ
∈
N
N
, i j
c
− D
sequence and taking into account of weak
σ
−
distributivity of
L
, we get
[ ] [ ]
ρ ρ
∞ =
∈
⎛ ⎞
− ≤ ∧ ∨
= ⎜
⎟ ⎝
⎠
∫ ∫
,
1 ,
, i
i i
a b a b
M fdx
M gdx
c
that is
[ ] [ ]
≤
∫ ∫
, ,
a b a b
M fdx
M gdx
. This concludes the proof.
Definition 3.5 Elementary Set: A set
[ ]
⊂ ,
a b R
which is union of finite cells is called
an elementary set.
Every elementary set can be segmented into non‐overlapping cells. If
and are elementary sets then
and
1
A
2
A ∪
1
A A
2 1
2
\ A
A
are also elementary sets. Integration
on elementary set can be constructed through the following theorem.
Teorema 3.6 : Let and be non‐overlapping intervals in and
. If
1
A
2
A R
= U
1
A A
A
2
∈ l
1
, , f
M A L
and , then
∈ l
2
, , f
M A L ∈
l , ,
f M A L
and
SEMNAS Matematika dan Pend. Matematika 2007 932
=
= +
∫ ∫
∫
U
1 2
1 2
A A A
A A
M fdx
M fdx
M fdx
Proof
: Let
∈ l
1
, , f
M A L
and
∈ l
2
, , f
M A L
. There exists two
− D
sequence
, ,
i j i j
a
and
, ,
i j i j
b
, such that for every
, we can find positive functions
ρ
∈
N
N
δ
1
and
δ
2
on respectively. Whenever
A
{ }
=
1
, P
A x
is
δ
1
‐fine M‐partition on and
1
A
{ }
=
2
, P
A x
is
δ
2
‐fine M‐partition on , we have
2
A
ρ
∞ =
− ≤ ∨
∑ ∫
l
1
1 ,
1 i
i i
A
P f x
I M
fdx a
and
ρ
α
∞ =
− ≤ ∨
∑ ∫
2
2 ,
1 i
i i
A
P f x
I M
fdx b
Let now
be such that,
δ
+
→ : A
R
{ }
δ δ
δ δ
δ
⎧ ∈
∉ ⎪
= ∈
⎨ ⎪
∈ ⎩
I
1 1
2 2
1 2
1 2
and and
min ,
∉
2 1
x if x
A x
A x
x if x
A x
x x
if x A
A A
for every
δ
‐fine M‐partition
{ }
= ,
P A x
on where
A =
U
1
P P
P
2
. Therefore, we
get
ρ ρ
ρ
∞ ∞
∞ =
= =
⎛ ⎞
− +
⎜ ⎟
⎜ ⎟
⎝ ⎠
≤ −
+ −
≤ ∨ + ∨
≤ ∨
∑ ∫
∫ ∑
∑ ∫
∫
l l
l
1 2
1 2
1 2
, ,
, 1
1 1
A A
A A
i i
i i
i i
i i
i
P f x
I M
fdx M
fdx P
f x I
M fdx
P f x
I M
fdx a
b c
where is a
= +
∀ ∈
, ,
,
2 ,
i j i j
i j
c a
b i j
N
− D
sequence.
Matematika 933
Using Theorem 3.6 and Definition 3.5 above, we can see immediately that
the following holds.
Corrolary 3.7 : Given an elementary set
. A function
is said to be McShane
integrable on , denoted by
⊂ A
R →
: f A
L A
∈ l
, , f
M A L
, if
∈ l
, ,
i
f M A L
for every i, where
and
=
= U
1 p
i i
A A
{ }
1 2
, ,...,
r
A A A
is any nonoverlapping intervals of . The McShane integral
of function on is
A f
A
=
⎡ ⎤
= ⎢
⎥ ⎢
⎥ ⎣
⎦
∑ ∫
∫
1
i
r i
A A
M fdx
M fdx
.
We now state version of the Cauchy criterion, where the proof is similar
to the one of Ansori 2007.
Theorem 3.8 : A function
[ ]
→ :
, f
a b L
is McShane integrable if and only if there exists
a
− D
sequence
, ,
i j i j
a
in such that, for every we can find a function
L
ρ
∈
N
N
[ ]
δ
+
→ :
, a b
R
and for every
δ
−
fine M‐ partition
[ ]
{ }
=
1
, ,
P a b x
and
[ ]
{ }
=
2
, ,
P a b x
on
[ ]
, a b
, we have
ρ
∞ =
− ≤
∑ ∑
l l
1 2
, 1
i i
i
P f x
I P
f x I
a ∨
.
We now provide a result about Hentock‐Kurzweil integrability on
subcells.
SEMNAS Matematika dan Pend. Matematika 2007 934
Theorem 3.9 : Let
[ ]
⊂ ,
a b R
. If
∈ l
, , f
M A L
, then
∈ l
, , f
M B L
, for every interval
[ ]
⊂ , B
a b
.
Proof : similar to that M‐integral with values in real spaces.
By virtue of Theorem 3.9, we define primitif function of McShane
integrable function on a cell
f
[ ]
⊂ ,
a b R
with respect to a volume
α
as follows.
Definition 3.10 : If
[ ]
∈ l
, , ,
f M a b L
and
[ ]
, I a b
is a collection of all subcells in
[ ]
, a b
, then
a function
[ ]
→ :
, F I a b
L
satisfying
=
∫
J
F J M
fdx
and
φ = 0 F
for every interval
∈ J
I A
is called Primitif of McShane integrable function on
f
[ ]
, I a b
.
References
Ansori, M., 2007. On The Henstock‐Kurzweil Integral for Value in Riesz Spaces
Defined on Euclidean Spaces, Proceeding of National Seminar FMIPA, UNY,
Yogyakarta. Boccuto,
A., 1998 , Differential and integral calculus in Riesz Spaces, Tatra Mountains
Math. Publ.,14,133‐323. Boccuto,
A and Riecan, B., 2004 , On The Henstock‐Kurzweil Integral for Riesz‐ Space
‐Valued Functions Defined on Unbounded Intervals, Chech. Math. Journal,
54,3, 591‐607. Pfeffer,
W.F., 1993 , The Riemann Approach to Integration, Cambridge University Press.
Riecan, B., 1989 , On the Kurzweil Integral for Functions with Values in
Ordered Spaces I, Acta Math. Univ. Comenian. 56‐57,75‐83.
Matematika 935
Riecan, B., 1992 , On Operator Valued Measures in Lattice ordered Groups,
Atti. Sem. Mat. Fis. Univ. Modena, 40, 151‐154.
Riecan, B and Neubrunn, T., 1997 , Integral, Measure and Ordering, Kluwer
Academic Publishers, Bratislava.
Riecan, B and Vrabelova, M., 1996 , On The Kurzweil integral for Functions
with Values in Ordered Spaces III, Tatra Mountains Math. Publ.,8, 93‐100.
Zaanen, A.A., 1997 , Introduction to Operator Theory in Riesz Spaces, Springer
Verlag.
SEMNAS Matematika dan Pend. Matematika 2007 936
Model Hazard Proporsional Semiparametrik dengan Hazard Dasar
Parametrik
Toha Saifudin dan Suliyanto
Jurusan Matematika, FMIPA Universitas Airlangga
Kampus C Jl. Mulyorejo Surabaya 60115 , Telp. 031‐5936501
Abstrak
Model hazard proporsional semiparametrik dalam penelitian ini diasumsikan mempunyai
bentuk
, |
X t
h t
h
Z X
Ψ =
dengan ,
Z merupakan vektor dari
peubah bebas, X peubah bebas,
merupakan vektor dari koefisien regresi ,
merupakan fungsi
hazard dasar, dan
} exp{
, X
T
λ
+ =
Ψ β
Z X
Z
1 px
1 px
t h
X λ
adalah fungsi smooth yang tidak diketahui. Tujuan dari tulisan ini adalah membahas
estimasi parameter model di atas jika digunakan hazard dasar parametrik dan diterapkan pada sampel
tersensor tipe I dengan menggunakan metode Generalized Profile Likelihood.
Kata kunci : hazard semiparametrik, fungsi smooth, Generalized Profile Likelihood
1. Pndahuluan
