EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1512-1520
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Pettis integrability in L1
E′[E] related to the truncation

Noureddine Sabiri1,∗, Mohamed Guessous1

1 Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan
II University of Casablanca, Casablanca, Morocco.

Abstract. We study the Pettis integrability in terms of truncation. We focus our study particu-
larly on space L1

E′ [E].

2020 Mathematics Subject Classifications: 28A20, 28A25, 40A05, 46G10
Key Words and Phrases: Convergence, Dual space, Gelfand integral, Pettis integral, Truncation

1. Introduction

Several authors studied the Pettis integrability of Banach space valued functions (see
for example [1],[10],[11],[13],[14],[12],[18] and references therein) and especially of dual Ba-
nach space valued functions ([2],[17],[19]). Similarly, the study of Pettis integrability for
multifunctions has been the focus of various papers (for example [9],[15] and [22]). In this
note, we are interested in Pettis integrability for scalarly integrable functions of L1

E′ [E].
Our study is based on the truncation technique that has been adopted in ([5],[6]) to state
some Komlós type theorems for Bochner integrable functions and in [16] to provide a
Komlós type theorem in L1

E′ [E]. It is well known that a strongly measurable and scalarly
integrable function f : Ω → E is Pettis integrable if and only if the set {⟨x′, f⟩ : ∥x′∥ ≤ 1}
is uniformly integrable in L1

R(µ) ([14] Theorem 5.2). We give a characterization of Pet-
tis integrability for scalarly integrable function (non-necessary strongly measurable) with
norm measurable function (Proposition 1) and, when E is a separable Banach space, we es-
tablish that a function f ∈ L1

E′ [E] is Pettis integrable if and only if its truncated function
1{∥f∥≤n}f is Pettis integrable for all n ≥ 1 (Corollary 1). We also give some criteria that
guarantee the Pettis integrability of the limit of a Pettis integrable L1

E′ [E]-convergent se-
quence. More precisely, we show that if a sequence of Pettis integrable functions bounded
in L1

E′ [E] converges weakly a.e. in E′ (resp. converges pointwise in L∞
R (µ)

⊗
E′′) to a

scalarly integrable function f , then f is Pettis integrable Theorem 2 (resp. Theorem 4).
It is important to note that a bounded scalarly integrable function is not in general Pettis
integrable, one can find some examples in [2],[19]. We note that the results in [16] will
play an important role for the development of this work and a version of Theorem 4 in
[16] with Pettis integrable functions is given (Theorem 6).

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4528

Email addresses: sabiri.noureddine@gmail.com (N. Sabiri), guessousjssous@yahoo.fr (M. Guessous)

https://www.ejpam.com 1512 © 2022 EJPAM All rights reserved.



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1513

2. Notations and Preliminaries

Let (Ω,F , µ) be a complete probability space, E a Banach space and E′ its topological
dual. The weak topology σ(E,E′) on E (resp. the weak* topology σ(E′, E) on E′) will be
referred to by the symbol ”w” (resp. ”w*”). A function f : Ω → E (resp f : Ω → E′) is
w-measurable (resp w*-measurable), if for any x′ ∈ E′, (resp x ∈ E) the function ⟨f, x′⟩ :
ω 7→ ⟨f(ω), x′⟩ (resp ⟨f, x⟩ : ω 7→ ⟨f(ω), x⟩) is measurable. Two functions f, g : Ω → E
(resp f, g : Ω → E′) are w-equivalent (resp w*-equivalent), if ⟨f, x′⟩ = ⟨g, x′⟩ µ − a.e. for
every x′ ∈ E′, (resp ⟨f, x⟩ = ⟨g, x⟩ µ− a.e. for every x ∈ E). A function f : Ω → E (resp
f : Ω → E′) is scalarly integrable (resp w*-scalarly integrable) if for every x′ ∈ E′ the
function ⟨f, x′⟩ (resp for every x ∈ E the function ⟨f, x⟩) is µ-integrable. If f : Ω → E is
scalarly integrable, then ([7] Lemma 1. p. 52) for every A ∈ F there exists x′′f (A) in E′′

such that, for every x′ ∈ E′

⟨x′′f (A), x′⟩ =
∫
A
⟨f, x′⟩ dµ,

the element x′′f (A) is called the Dunford integral of f over A and denoted by (D)−
∫
A fdµ.

By definition, f is Pettis integrable if (D) −
∫
A fdµ ∈ E for all A ∈ F and we write

(P ) −
∫
A fdµ instead of (D) −

∫
A fdµ. Also, ([7] p. 53) if f : Ω → E′ is w*-scalarly

integrable then for every A ∈ F there exists x′f (A) in E′ such that, for every x ∈ E

⟨x′f (A), x⟩ =
∫
A
⟨f, x⟩ dµ,

the element x′f (A) is called the weak* integral (or Gelfand integral) of f over A and

denoted by (w∗) −
∫
A fdµ. A sequence (fn) of E-valued scalarly integrable functions

converges pointwise on L∞
R (µ)

⊗
E′ to an E-valued scalarly integrable function f if

∀h ∈ L∞
R (µ),∀x′ ∈ E′,

∫
Ω
h⟨fn, x′⟩ dµ →

∫
Ω
h⟨f, x′⟩ dµ,

or equivalently ([8] Theorem 7. p. 291) for every x′ ∈ E′, the sequence (⟨fn, x′⟩)n is
bounded in L1

R(µ) and

∀A ∈ F ,

∫
A
⟨fn, x′⟩ dµ →

∫
A
⟨f, x′⟩ dµ.

Let P 1
E(µ) denote the (quotient) space of Pettis integrable E-valued functions. The weak

topology on P 1
E(µ) is the weak topology induced by the duality (P 1

E(µ), L
∞
R (µ)

⊗
E′). If

E is separable and f : Ω → E′ is w*-measurable, the function ∥f(.)∥ is measurable [20]
however, this is not always the case if E is a general Banach space ([14] Example 3.3). With
E being separable, the Banach space (L1

E′ [E] , N1) ([3],[21],[16]) is simply the (quotient)
space of w*-scalarly integrable functions f : Ω → E′ such that ∥f(.)∥ is µ-integrable, and

N1(f) =

∫
Ω
∥f(ω)∥ dµ(ω), f ∈ L1

E′ [E] .



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1514

Finally, we recall that a set H of L1
R(µ) is uniformly integrable (briefly UI) if it is bounded

and

lim
µ(A)→0

sup
f∈H

∫
A
|f | dµ = 0.

A set K of L1
E′ [E] is UI [16] if the set {∥f(.)∥ : f ∈ K} is UI in L1

R(µ), and we say that a
set H of E-valued scalarly integrable functions is scalarly uniformly integrable briefly SUI
(resp w-scalarly uniformly integrable briefly WSUI), if the set {⟨x′, f⟩ : ∥x′∥ ≤ 1, f ∈ H}
(resp for each x′ ∈ E′, the set {⟨x′, f⟩ : f ∈ H}) is UI in L1

R(µ).

3. Pettis integrability and truncation

By ([10] p.82), if f : Ω → E is Pettis integrable then {f} is SUI and the converse
remains true if f is strongly measurable ([14] Theorem 5.2). For the instance of L1

E′ [E],
we give some characterizations of the Pettis integrability by the mean of the associated
truncated functions. Our work build on the following ([4], Theorem 3.1):

Theorem 1. Let E be a Banach space, (fn) a sequence of E-valued Pettis integrable
functions and f : Ω → E a scalarly integrable function satisfying:

(i) {f} is SUI,

(ii) (fn) converges pointwise on L∞
R (µ)

⊗
E′ to f .

Then f is Pettis integrable.

The next lemma is useful.

Lemma 1. If f : Ω → E is scalarly integrable and ∥f(.)∥ is measurable, then the sequence
(1{∥f∥≤n}f)n converges pointwise on L∞

R (µ)
⊗

E′ to f .

Proof. Let h ∈ L∞
R (µ) and x′ ∈ E′. We have

h(ω)⟨1{∥f∥≤n}f(ω), x
′⟩ → h(ω)⟨f(ω), x′⟩ ∀ω ∈ Ω,

and
|h(ω)⟨1{∥f∥≤n}f(ω), x

′⟩| ≤ ∥h∥∞|⟨f(ω), x′⟩| a.e.,

then by the Lebesgue dominated convergence theorem∫
Ω
|⟨h(ω)1{∥f∥≤n}f(ω)− f(ω), x′⟩|dµ(ω) → 0,

and therefore ∫
Ω
h(ω)⟨1{∥f∥≤n}f(ω), x

′⟩dµ(ω) →
∫
Ω
h(ω)⟨f(ω), x′⟩dµ(ω).



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1515

Proposition 1. If f : Ω → E is scalarly integrable and ∥f(.)∥ is measurable, then f is
Pettis integrable if and only if

(i) {f} is SUI, and

(ii) 1{∥f∥≤n}f is Pettis integrable for all n ≥ 1.

Proof. If f is Pettis integrable then {f} is SUI and 1{∥f∥≤n}f is Pettis integrable
∀n ≥ 1. The converse follows from Theorem 1 and Lemma 1.

The above result gives a characterization of Pettis integrability for scalarly integrable
function with measurable norm function (compare with Theorem 5.2 in [14]) and it can be
seen as a generalization for the case of strongly measurable functions since, if f : Ω → E is
strongly measurable then ∥f(.)∥ is measurable and hence 1{∥f∥≤n}f is Bochner then Pettis
integrable. We obtain the following characterization of Pettis integrability in L1

E′ [E].

Corollary 1. Let E be a separable Banach space and f ∈ L1
E′ [E]. Then f is Pettis

integrable iff 1{∥f∥≤n}f is Pettis integrable for all n ≥ 1.

Proof. As E is separable then ∥f(.)∥ is measurable. The direct implication is immediate
we show the converse. For every x′′ ∈ E′′ the function ⟨f(.), x′′⟩ is measurable a simple
limit of (⟨1{∥f∥≤n}f(.), x

′⟩)n. For all ω ∈ Ω and x′′ ∈ BE′′ , we have |⟨f(ω), x′′⟩| ≤ ∥f(ω)∥.
As ∥f(.)∥ ∈ L1

R(µ) then {f} is SUI. Therefore we apply Proposition 1.

From now, we suppose that E is separable. If (fn) is a convergent sequence of Pettis
integrable functions of L1

E′ [E], when does (fn) have a Pettis integrable limit? Here the
convergence is taken in the sense of weak convergence a.e. or the pointwise convergence
on L∞

R (µ)
⊗

E′′. The following result is an analogue of Vitali’s convergence theorem for
Pettis integrable functions.

Lemma 2. Let f ∈ L1
E′ [E] be a scalarly integrable function. Suppose that there exists a

sequence of Pettis integrable functions (fn) such that

(i) (fn) is WSUI, and

(ii) for each x′′ ∈ E′′, limn→∞⟨fn, x′′⟩ = ⟨f, x′′⟩ a.e.

Then f is Pettis integrable and (fn) converges weakly to f in P 1
E′(µ).

Proof. As ∥f(.)∥ is integrable then {f} is SUI. We apply Theorem 1 and Vitali’s
theorem in L1

R(µ).

Theorem 2. Let (fn)n∈N a bounded sequence in L1
E′ [E].

If (fn) w*-converges a.e. to a function f : Ω → E′ then f ∈ L1
E′ [E].

If fn is Pettis integrable for all n and (fn) w-converges a.e. to f , then f is Pettis integrable.



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1516

Proof. As (fn(ω))n w*-converges a.e. to f(ω) we have

∥f(ω)∥ ≤ lim inf
n

∥fn(ω)∥ a.e.

By Fatou’s lemma and the boundedness of (fn) in L1
E′ [E] we get∫

Ω
∥f∥dµ ≤ lim inf

n

∫
Ω
∥fn∥dµ < ∞,

thus f ∈ L1
E′ [E].

Now suppose that fn is Pettis integrable for all n and (fn) w-converges a.e. to f . Then f is
w-measurable with ∥f(.)∥ is integrable, so that f is scalarly integrable. By Lemma 2 in [16]
there exists a subsequence (gn) of (fn) such that (1{∥gn∥<n}gn) is UI and (gn−1{∥gn∥<n}gn)
converges a.e. to 0 in E′, hence (1{∥gn∥<n}gn) is WSUI and weakly converges a.e. to f . It
remains to use Lemma 2 to conclude that f is Pettis integrable.

Now we give a criterion of the σ(L1
E′ [E] , L∞

R (µ)
⊗

E)-compactness for L1
E′ [E]-bounded

subsets.

Theorem 3. Let H a bounded subset of L1
E′ [E]. Then H is σ(L1

E′ [E] , L∞
R (µ)

⊗
E)-

sequentially relatively compact if and only if for each x ∈ E the set Hx = {⟨f, x⟩ : f ∈ H}
is UI in L1

R(µ).

Proof. If H is σ(L1
E′ [E] , L∞

R (µ)
⊗

E)-sequentially relatively compact then for each
x ∈ E, Hx is σ(L1

E′ [E] , L∞
R (µ))-sequentially relatively compact and equivalently is UI

in L1
R(µ). Conversely, let (fn) a sequence of H. By Theorem 2 in [16] there exists a

subsequence (f ′
n) of (fn) and a function f ∈ L1

E′ [E] such that, for every subsequence (hn)
of (f ′

n)

(⟨ 1
n

n∑
i=1

hi(ω)) w*-converges a.e. to f(ω).

For every x ∈ E, the sequence (⟨ 1n
n∑

i=1
hi, x⟩)n is UI in L1

R(µ) since Hx it is, then by the

Vitali’s theorem in L1
R(µ)

∀A ∈ F ,

∫
A
⟨ 1
n

n∑
i=1

hi, x⟩dµ →
∫
A
⟨f, x⟩ dµ. (1)

As (1) is valid for every subsequence (hn) of (f ′
n), by an elementary property of Cesàro

convergence in R we get

∀A ∈ F ,∀x ∈ E

∫
A
⟨f ′

n, x⟩dµ →
∫
A
⟨f, x⟩ dµ.

It follows by the boundedness of (fn) in L1
E′ [E] that

∀h ∈ L∞
R (µ), ∀x ∈ E,

∫
Ω
h⟨f ′

n, x⟩dµ →
∫
Ω
h⟨f, x⟩ dµ.



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1517

Thus (fn) is σ(L
1
E′ [E] , L∞

R (µ)
⊗

E)-sequentially relatively compact.

The next result show that if a sequence of Pettis integrable functions bounded in
L1
E′ [E] converges pointwise in L∞

R (µ)
⊗

E′′ to a scalarly integrable function f , then f is
Pettis integrable.

Theorem 4. Let f : Ω → E′ and (fn) a bounded sequence of L1
E′ [E].

(1) If f is w*-scalarly integrable and

∀A ∈ F ,∀x ∈ E,

∫
A
⟨fn, x⟩ dµ →

∫
A
⟨f, x⟩ dµ, (2)

then f ∈ L1
E′ [E].

(2) If f is scalarly integrable, fn is Pettis integrable for all n and

∀A ∈ F ,∀x′′ ∈ E′′,

∫
A
⟨fn, x′′⟩ dµ →

∫
A
⟨f, x′′⟩ dµ, (3)

then f is Pettis integrable.

Proof. (1) By the Vitali-Hahn-Saks theorem ([7] Corollary I.4.10) for each x ∈ E
the sequence (⟨fn, x⟩)n is UI in L1

R(µ). Applying Theorem 3 to (fn) we have that (fn)
is σ(L1

E′ [E] , L∞
R (µ)

⊗
E)-sequentially relatively compact, so there exists a subsequence

(f ′
n) converging σ(L1

E′ [E] , L∞
R (µ)

⊗
E) to a g ∈ L1

E′ [E]. Then we have

∀A ∈ F ,∀x ∈ E,

∫
A
⟨f ′

n, x⟩ dµ →
∫
A
⟨g, x⟩ dµ. (4)

By (2) and (4) we get

∀A ∈ F ,∀x ∈ E,

∫
A
⟨g, x⟩ dµ =

∫
A
⟨f, x⟩ dµ,

hence
∀x ∈ E, ⟨g, x⟩ = ⟨f, x⟩. a.e.

It follows by the separability of E that ∥g∥ = ∥f∥ a.e. and therefore f ∈ L1
E′ [E].

(2) Now suppose that f is scalarly integrable, fn is Pettis integrable for each n and (3) is
satisfied and let us prove that f is Pettis integrable. By Theorem 1 it is enough to check
that {f} is SUI, which is the case since ∥f(.)∥ is integrable.

The next result is an immediate application of the above theorem.

Corollary 2. The subset of L1
E′ [E] of Pettis integrable functions is norm closed.



N. Sabiri, M. Guessous / Eur. J. Pure Appl. Math, 15 (4) (2022), 1512-1520 1518

Proof. Let (fn) a norm convergent sequence of Pettis integrable functions of L1
E′ [E]

and f its limit in L1
E′ [E], there exists a subsequence (f ′

n) of (fn) such that

lim
n

∥f ′
n(ω)− f(ω)∥ = 0 a.e.

So f is w-measurable and

∀A ∈ F ,∀x′′ ∈ E′′,

∫
A
⟨fn, x′′⟩ dµ →

∫
A
⟨f, x′′⟩ dµ.

By Theorem 4 (2) f is Pettis integrable.

By combining Theorem 2 and Theorem 4 we have the following.

Theorem 5. Let (fn) a bounded sequence of L1
E′ [E]. Suppose that the following hold:

(1) fn w*-converges a.e. to a function f ,

(2) fn is Pettis integrable for each n,

(3) for each k ∈ N∗ there is a scalarly integrable function vk such that

∀A ∈ F , ∀x′′ ∈ E′′,

∫
A
⟨1{∥fn∥≤k}fn, x

′′⟩ dµ →
∫
A
⟨vk, x′′⟩ dµ. (5)

Then f is Pettis integrable.

Proof. By (1) and Theorem 2 we get f ∈ L1
E′ [E], so by Corollary 1 we have to prove

that 1{∥f∥≤k}f is Pettis integrable for every k ∈ N∗. Fix k ∈ N∗ and applying Theorem 4
(2) to vk and (1{∥fn∥≤k}fn)n we get that vk is Pettis integrable. As (1{∥fn∥≤k}fn)n is WSUI
and by (1) is w*-converges a.e. to 1{∥f∥≤k}f , it follows by the Vitali’s theorem in L1

R(µ)
that

∀A ∈ F ,∀x ∈ E,

∫
A
⟨1{∥fn∥≤k}fn, x⟩ dµ →

∫
A
⟨1{∥f∥≤k}f, x⟩ dµ. (6)

By (5) and (6) we get

∀x ∈ E, ⟨1{∥f∥≤k}, x⟩ = ⟨vk, x⟩ a.e.

Being E separable, it follows that vk = 1{∥f∥≤k}f a.e. and therefore 1{∥f∥≤k}f is Pettis
integrable.

We finish this work by the following version of Theorem 4 in [16] with Pettis integrable
functions. Recall that Rwc(E′) denoted the set of nonempty convex ball weakly compact
subsets of E′.

Theorem 6. Let (fn) be a bounded sequence in L1
E′ [E]. Suppose that fn is Pettis integrable

for all n ∈ N and there exist a Rwc(E′)-valued multifunction Γ such that fn(ω) ∈ Γ(ω) for
a.e. ω ∈ Ω and for all n ∈ N. Then there exists a Pettis integrable function f ∈ L1

E′ [E]
and a subsequence (gn) of (fn) such for every subsequence (hn) of (gn) the following holds



REFERENCES 1519

(j) ( 1n

n∑
i=1

hi) w-converges a.e. to f.

(jj) (1{∥hn∥<n}hn) converges σ(L1
E′ [E] , (L1

E′ [E])′) (weakly) to f in L1
E′ [E] and (hn −

1{∥hn∥<n}hn) converges a.e. to 0 in E′.

Proof. By Theorem 4 in [16] there exists a function f ∈ L1
E′ [E] and a subsequence

(gn) of (fn) such that (j) and (jj) hold. Now since ( 1n

n∑
i=1

hi) is bounded in L1
E′ [E] and

weak converges a.e. to f , it follows by Theorem 2 that f is Pettis integrable.

Acknowledgements

The authors wish to thank the referees for their constructive critique of the first draft.

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