Ann. Funct. Anal. 4 (2013), no. 1, 1–10
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
THE BOCHNER INTEGRAL FOR MEASURABLE SECTIONS
AND ITS PROPERTIES
INOMJON GANIEV∗ AND GHARIB S. MAHMUOD
Communicated by D. H. Leung
Abstract. In the present paper we introduce the notion Bochner integral for
measurable sections and study some properties such integrals. Given necessary
and successfully condition for integrability of a measurable section. Dominated
convergence theorem and analogue of Hille’s theorem are proved.
1. Introduction
Bochner integral is used in many mathematics field, such as probability theory,
functional analysis, differential equations in vector spaces, theory of semigroup of
linear operators and so on. The integral of Banach valued function was introduced
by Bochner [1] and Pettis [2]. Integration of function with values in locally convex
spaces considered by Phillips [3] and Rikkard [4]. The Bochner integral of Banach
valued functions and its applications are given in many books and monographs,
for example, in Hille and Fillips [5], Yosida [6] Bogachev [7], Vakhania et al [8],
Schwabik [9]. The Bochner integral is used in Arendt et al [10] to solve different
problems of analysis. For example in [11] it is used to study geometry of Banach
spaces, in [12] it is used to study semigroup of linear operators in Banach spaces.
It is known that the theory of Banach bundles stemming from paper [13], where
it was showed that such a theory has vast applications in analysis. For another
applications of the measurable Banach bundles, we refer the reader [14, 16, 17].
In [14] and [15] Gutman introduced the notion measurable section and showed
properties of measurable sections obtained by means of a measurability structure
and proved that every Banach– Kantorovich space over ring measurable functions
Date: Received: 9 April 2012; Accepted: 27 June 2012.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 46G10; Secondary 46G12, 46E40.
Key words and phrases. Measurable bundle; measurable section; integral Bochner.
1
2
I. GANIEV, G.S. MAHMUOD
is linearly isometric to the space of measurable sections of a measurable Banach
bundle.
In present paper we generalize the notion Bochner integral for measurable
sections and study properties of such integrals.
2. Preliminaries
Let (Ω, Σ, λ) be the space with finite measure, L0 = L0 (Ω) algebra of classes
measurable functions on (Ω, Σ, λ). Lp (Ω) be Banach space of measurable funcR
1
tions integrable with degree p, p ≥ 1, with norm kf kp = ( |f (ω)|p dλ) p .
Ω
An ideal space on (Ω, Σ, λ) is a linear subset E in L0 such that
(x ∈ L0 , y ∈ E; |x| ≤ |y|) ⇒ (x ∈ E)
i.e., with every function the set E contains its modulus and each function with
smaller modulus. The basic examples are L0 , Lp (Ω), L∞ , Orlicz and Marsinkevicz
spaces. Denote by E + the cone of positive elements or the positive cone of an
ideal space E:
E + = {x ∈ E : x ≥ 0}.
A sequence bn is said to be order convergent (or o-convergent) to b if there is a
(o)
sequence an in E satisfying an ↓ 0 and |bn − b| ≤ an for all n. We write bn → b
or b = (o) − lim bn denote order convergence.
n
We will consider vector spaces F over field real numbers R.
Definition 2.1. [18] A map k · k : F −→ E is called an E– valued norm on F ,
if for any x, y ∈ F, λ ∈ R it satisfies the following conditions:
1) kxk ≥ 0; kxk = 0 ⇐⇒ x = 0;
2) kλxk = |λ|kxk;
3) kx + yk ≤ kxk + kyk.
A pair (F, k · k) is called lattice-normed space (LNS) over E.
A LNS F is said to be d-decomposable, if for any x ∈ F and for any decomposition kxk = f + g to sum disjoint elements there exists such y, z ∈ F , that
x = y + z kxk = f, kzk = g.
A net {xα } in F is called (bo)-convergent to x ∈ F , if the net {kxα − xk} is
(o)-convergent to zero in E.
A lattice normed space is called (bo)-complete if every (bo)-fundamental net
is (bo)-convergent in it. A Banach–Kantorovich space (BKS) over E is a (bo)complete d-decomposable lattice normed space over E. It is well known [18] that
every Banach–Kantorovich space F over E admits an E-module structure such
that ||λx|| = |λ|||x|| for every x ∈ F, λ ∈ E.
Let X be a mapping, which maps every point ω ∈ Ω to some Banach space
(X(ω), k · kX(ω) ). In what follows, we assume that X(ω) 6= {0} for all ω ∈ Ω. A
function u is said to be a section of X, if it is defined almost everywhere in Ω and
takes its value u(ω) ∈ X(ω) for ω ∈ dom(u), where ω ∈ dom(u) is the domain of
u.
Let L be some set of sections.
BOCHNER INTEGRAL FOR MEASURABLE SECTIONS
3
Definition 2.2. [14]. A pair (X, L) is said to be a measurable bundle of Banach
spaces over Ω if
1. λ1 c1 + λ2 c2 ∈ L for all λ1 , λ2 ∈ R and c1 , c2 ∈ L, where λ1 c1 + λ2 c2 : ω ∈
dom(c1 ) ∩ dom(c2 ) → λ1 c1 (ω) + λ2 c2 (ω);
2. the function ||c|| : ω ∈ dom(c) → ||c(ω)||X(ω) is measurable for all c ∈ L;
3. for every ω ∈ Ω the set {c(ω) : c ∈ L, ω ∈ dom(c)} is dense in X(ω);
A section s is said to be step, if there are ci ∈ L, Ai ∈ Σ, i = 1, n such that
n
P
s(ω) =
χAi (ω)ci (ω) for almost all ω ∈ Ω.
i=1
A section u is called measurable if there is a sequence {sn } of step sections such
that sn (ω) → u(ω) almost everywhere on Ω.
The set of all measurable sections is denoted by M (Ω, X), and L0 (Ω, X) denotes
the factorization of this set with respect to equality everywhere. We denote by
û the class from L0 (Ω, X) containing a section u ∈ M (Ω, X), and by ||û|| the
element of L0 containing the function ||u(ω)||X(ω) .
It is known [14, 15] that L0 (Ω, X) is a BKS over L0 .
3. The Bochner integral for measurable sections and its
properies
Let s be a step section and mi = sup kci (ω)kX(ω) < ∞ for any i = 1, 2, ..., n
ω∈dom(ci )
R
then ks(ω)kX(ω) dλ < ∞. Actually, as kci (ω)kX(ω) < mi we have
Ω
ks(ω)kX(ω) =
n
X
χAi (ω)kci (ω)kX(ω) ≤
n
X
mi χAi (ω).
i=1
i=1
Therefore
Z
ks(ω)kX(ω) dλ ≤
n
X
mi λ(Ai ) < ∞.
i=1
Ω
We define the integral of step section by measure λ with equality
Z
n
X
s(ω)dλ =
ci (ω)λ(Ai ).
i=1
Ω
From this definition it follows that
Z
Z
n
n
X
X
s(ω)dλ
=
ci (ω)λ(Ai )
≤
kci (ω)kX(ω) λ(Ai ) = ks(ω)kX(ω) dλ.
i=1
i=1
X(ω)
Ω
Ω
X(ω)
Definition 3.1. The measurable section u is said to be integrable by Bochner, if
there exists a sequence step sections sn such that
Z
lim
ksn (ω) − u(ω)kX(ω) dλ = 0
n→∞
Ω
4
I. GANIEV, G.S. MAHMUOD
In this case the integral
R
udλ for every A ∈ Σ defined with equality
A
Z
Z
udλ = lim
sn dλ
n→∞
A
(3.1)
A
By analogy of Banach valued case, it can be proved, that definition is correct,
i.e. (3.1) independent from choosing the sequence step sections.
Theorem 3.2. If u is a measurable section
R such that u(ω) = 0 for almost all
ω ∈ Ω then u is integrable by Bochner and u(ω)dλ = 0.
Ω
Proof. The sequence of step sections from Definition 3.1 can be chosen as sections
which are identically zero.
Corollary 3.3. If the section u is Bochner integrable and v is a section
R such that
u(ω) = v(ω) for almost all ω ∈ Ω then v is Bochner integrable and u(ω)dλ =
Ω
R
v(ω)dλ
Ω
Proof. As u = u − v + v and u − v is Bochner integrable by Theorem 3.2, we get
the statement immediately.
Proposition 3.4. A countably valued measurable section u of the form
u(ω) =
∞
X
ci (ω)χAi (ω), ci ∈ L, Ai ∈ Σ, Ai ∩ Aj = Ø
i=1
is Bochner integrable if
∞
X
kci (ω)kX(ω) λ(Ai ) < ∞.
i=1
Proof. For any n ∈ N define sections sn (ω) =
n
P
ci (ω)χAi (ω). Then lim sn (ω) =
n→∞
i=1
u(ω) a.e. on Ω. For a.e. on Ω and k < n the following equality is valid equality
n
X
ksk (ω) − sn (ω)kX(ω) = k
ci (ω)χAi (ω)kX(ω) .
i=k+1
As
k
n
X
i=k+1
n
X
ci (ω)χAi (ω)kX(ω) =
kci (ω)kX(ω) χAi (ω)
i=k+1
we have
Z
ksk (ω) − sn (ω)kX(ω) dλ =
Ω
n
X
i=k+1
kci (ω)kX(ω) λ(Ai ).
BOCHNER INTEGRAL FOR MEASURABLE SECTIONS
Since
∞
P
∞
P
kci (ω)kX(ω) λ(Ai ) < ∞, we get that
5
ci (ω)χAi (ω) is convergent in X(ω)
i=1
i=1
to u(ω). Then by the definition of Bochner integral we have
Z
∞
X
u(ω)dλ =
ci (ω)λ(Ai )
i=1
Ω
and
Z
ku(ω)kX(ω) dλ =
∞
X
kci (ω)kX(ω) λ(Ai ).
i=1
Ω
Corollary 3.5. A countably valued measurable section u for which ku(ω)kX(ω) ≤
g(ω) a.e. with g ∈ L1 (Ω) is Bochner integrable.
n
P
ci (ω)χAi (ω) we get
Proof. Using the sequence sn (ω) =
i=1
Z
Z
ksn (ω)kX(ω) dλ ≤ g(ω)dλ < ∞
Ω
Ω
for any n ∈ N. Then by Proposition 3.4 u is Bochner integrable.
Theorem 3.6. A measurable section u is integrable by Bochner if and only if
Z
ku(ω)kX(ω) dλ < ∞.
Ω
Proof. Let the measurable sectionR u be integrableR by Bochner and sn be a sequence of step sections such that u(ω)dλ = lim sn (ω)dλ. Then
Ω
Z
n→∞ Ω
Z
Z
ku(ω)kX(ω) dλ ≤
Ω
ksn (ω) − u(ω)kX(ω) dλ +
Ω
ksn (ω)kX(ω) dλ < ∞.
Ω
On the contrary, let u be measurable section and
Z
ku(ω)kX(ω) dλ < ∞.
Ω
By [15, Proposition 4.1.8 (2)] there is a sequence of measurable sections gn in
form
∞
X
(n)
(n)
(n)
(n)
(n)
ci (ω)χA(n) (ω), ci ∈ L, Ai ∈ Σ, Ai ∩ Aj = Ø
i=1
i
when i 6= j, such, that kgn (ω) − u(ω)kX(ω) < n1 for almost all ω ∈ Ω. Then
kgn (ω)kX(ω) ≤ ku(ω)kX(ω) + n1 .
For any n we will choose pn ∈ N such, that
Z
λ(Ω)
.
kgn (ω)kX(ω) dλ <
n
∞
S
n=pn +1
(n)
Ai
6
I. GANIEV, G.S. MAHMUOD
Put
sn (ω) =
pn
X
(n)
ci (ω)χA(n) (ω).
i
i=1
Then sn is a step section and
Z
Z
ksn (ω) − u(ω)kX(ω) dλ ≤ kgn (ω) − u(ω)kX(ω) dλ+
Ω
Ω
Z
ksn (ω) − gn (ω)kX(ω) dλ ≤
+
λ(Ω) λ(Ω)
2λ(Ω)
+
=
n
n
n
Ω
So the section u is integrable by Bochner.
Corollary 3.7. A measurable section u for which ku(ω)kX(ω) ≤ g(ω) a.e. with
g ∈ L1 (Ω) is Bochner integrable.
The following simple properties of integral Bochner are hold:
Theorem
3.8. If a section u is integrable by Bochner, then
R
R
(1) ≤ ku(ω)kX(ω) dλ for all A ∈ Σ;
u(ω)dλ
A
A
X(ω)
R
(2)
lim
u(ω)dλ = 0;
λ(A)→0 A
(3) If c ∈ L, f ∈ L1 (Ω) and
sup kc(ω)kX(ω) < ∞ then cf is integrable by
ω∈dom(c)
Bochner and
R
c(ω)f (ω)dλ = c(ω)
Ω
R
f (ω)dλ.
Ω
R
R
R
sn (ω)dλ
Proof. (1). u(ω)dλ
=
lim
≤
lim
ksn (ω)kX(ω) dλ =
n→∞ Ω
n→∞ Ω
A
X(ω)
X(ω)
R
= ku(ω)kX(ω) dλ.
Ω
R
(2). As lim
ku(ω)kX(ω) dλ = 0 from (1) follows, that
λ(A)→0 A
Z
Z
u(ω)dλ
≤ lim
ku(ω)kX(ω) dλ = 0
λ(A)→0
A
A
X(ω)
R
i.e. lim
u(ω)dλ = 0.
λ(A)→0 A
(3). Let f be a simple function from L1 (Ω) i.e. f (ω) =
n
P
λi χAi (ω), where λi ∈
i=1
R, Ai ∈ Σ, i = 1, n, Ai ∩ Aj = ∅ when i 6= j. Then c(ω)f (ω) =
n
P
i=1
c(ω)λi χAi (ω) is
n
n
R
P
P
step section and by definition c(ω)f (ω)dλ =
c(ω)λi λ(Ai ) = c(ω) λi λ(Ai ) =
i=1
i=1
Ω
R
c(ω) f (ω)dλ.
Ω
BOCHNER INTEGRAL FOR MEASURABLE SECTIONS
7
Now let f ∈ L1 (Ω). Then there exists a sequence fn of simple functions such,
that
Z
Z
f (ω)dλ = lim
fn (ω)dλ.
n→∞
Ω
Ω
Hence
R
R
R
R
c(ω)f (ω)dλ = lim c(ω)fn (ω)dλ = c(ω) lim fn (ω)dλ = c(ω) f (ω)dλ.
n→∞ Ω
Ω
n→∞ Ω
Ω
Theorem 3.9. (Dominated convergence) Let un be a sequence of sections, each of
which is Bochner integrable and there exist a section u and an integrable function
g such
1) lim un (ω) = u(ω) for almost all ω ∈ Ω;
n→∞
2)kun (ω)kX(ω) ≤ |g(ω)| for almost all ωR ∈ Ω.
Then u is Bochner integrable and lim kun (ω) − u(ω)kX(ω) dλ = 0. In particn→∞ Ω
ular we have that
Z
Z
un (ω)dλ =
lim
n→∞
Ω
u(ω)dλ.
Ω
Proof. Since kun (ω)kX(ω) → ku(ω)kX(ω) almost everywhere on Ω, we get that
ku(ω)kX(ω) ≤ |g(ω)|. Therefore, kun (ω) − u(ω)kX(ω) ≤ 2|g(ω)| for almost all
ω ∈ Ω and the result follows from the scalar dominated convergence theorem. Let L be the set of sections from Definition 2.2.
Theorem 3.10. (Hille) Let Tω : X(ω) → X(ω), ω ∈ Ω is a family of bounded
linear operators, such that Tω (c(ω)) ∈ L for any c ∈ L and kTω k ≤ 1. If section
u is Bochner integrable, then the section Tω (u(ω)) is Bochner integrable and


Z
Z
Tω (u(ω))dλ = Tω  u(ω)dλ
Ω
Ω
Proof. Let s be simple section. Then

T (c (ω)),


 ω 1
Tω (c2 (ω)),
Tω (s(ω)) =
...............


 T (c (ω)),
ω
n
if ω ∈ A1 ;
if ω ∈ A2 ;
...............
if ω ∈ An .
Since Tω (ci (ω)) ∈ L, we have that Tω (s(ω)) is simple section. Therefore, Tω (s(ω))
is Bochner integrable and


!
Z
Z
n
n
X
X
Tω s(ω)dλ =
Tω (ci (ω))λ(Ai ) = Tω
ci (ω)λ(Ai ) = Tω  s(ω)dλ
Ω
i=1
i=1
Ω
for almost all ω ∈ Ω.
Now let section u be Bochner integrable. Then there exists a sequence simple sections
sn such that ksn (ω) − u(ω)kX(ω) → 0 for almost all ω ∈ Ω and
R
lim ksn (ω) − u(ω)kX(ω) dλ = 0.
n→∞ Ω
8
I. GANIEV, G.S. MAHMUOD
Since kTω k ≤ 1 we have that
kTω (sn (ω)) − Tω (u(ω))kX(ω) ≤ ksn (ω) − u(ω)kX(ω)
and
kTω (sn (ω)) − Tω (u(ω))kX(ω) → 0
for almost all ω ∈ Ω. From equality
Z
Z
kTω (sn (ω)) − Tω (u(ω))kX(ω) dλ ≤ ksn (ω) − u(ω)kX(ω) dλ
Ω
Ω
we obtain that
Z
kTω (sn (ω)) − Tω (u(ω))kX(ω) dλ = 0.
lim
n→∞
Ω
As Tω (sn (ω)) is sequence of simple sections, Tω (u(ω)) is Bochner integrable. Then


Z
Z
Z
Tω (u(ω))dλ = lim
Tω (sn (ω))dλ = lim Tω  sn (ω)dλ =
n→∞
Ω
n→∞
Ω

Ω

Tω  lim


Z
Z
u(ω)dλ .
sn (ω)dλ = Tω 
n→∞
Ω
Ω
1. We define by Lp (Ω, X) all classes measurable sections for which
R Let p ≥
ku(ω)kpX(ω) dλ < ∞, i.e.
Ω
Lp (Ω, X) = {u ∈ L0 (Ω, X) : kukp ∈ L1 (Ω)}.
Then Lp (Ω, X) is a Banach-Kantorovich space over Lp (Ω) (see [14]) and according
to [18, Theorem 7.13 (2)] it is Banach space with respect to the mixed norm

 p1
Z
 ku(ω)kp dλ .
kukp = X(ω)
kuk =
Ω
Let ϕ1 , ϕ2 , ..., ϕn ∈ Lp and c1 , c2 , ..., cn ∈ L, mi =
sup
kci (ω)kX(ω) < ∞ for
ω∈dom(ci )
any i = 1, 2, ..., n. Then we can define a section u : Ω → X(ω) in Lp (Ω, X) by setn
N
P
ϕi (ω)ci (ω) for almost all ω ∈ Ω. We will denote by Lp (Ω) L =
ting u(ω) =
i=1
{u : u(ω) =
n
P
ϕi (ω)ci (ω)} subspace of Lp (Ω, X).
i=1
Theorem 3.11. The subspace Lp (Ω)
N
L is dense in Lp (Ω, X).
BOCHNER INTEGRAL FOR MEASURABLE SECTIONS
9
N
Proof. Let u ∈ Lp (Ω, X) and sn (ω) be the sequence step sections from Lp L
such that sn (ω) → u(ω) a.e. on Ω. Then ksn (ω)kX(ω) → ku(ω)kX(ω) almost
everywhere on Ω. Let An = {ω : ksn (ω)kX(ω) < 2ku(ω)kX(ω) }. If we set gn (ω) =
sn (ω)χAn (ω) we have that gn (ω) → u(ω) a.e. on Ω and
sup kgn (ω) − u(ω)kX(ω) ≤ sup kgn (ω)kX(ω) + ku(ω)kX(ω) ≤ 3ku(ω)kX(ω) .
n
n
By dominated convergence we have that
Z
kgn (ω) − u(ω)kpX(ω) dλ → 0
Ω
and of course gn ∈ Lp (Ω)
N
L.
Acknowledgement. This work has been supported by IIUM-RMC through
projects EDW B 11-185-0663 and EDW B 11-199-0677.
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10
I. GANIEV, G.S. MAHMUOD
Department of Science in Engineering, Faculty of Engineering, International
Islamic University Malaysia, P.O. Box 10, 50728 Kuala Lumpur, Malaysia.
E-mail address: inam@iium.edu.my
E-mail address: gharib@iium.edu.my