Disjoint sequence methods from the theory of Riesz spaces are used to study measures of weak non-compactness in
L
1
(
μ
)
L_{1}(\mu )
-spaces. A principal new result of the present paper is the following: Let
E
E
be an abstract
M
M
-space. Then
ω
(
B
)
a
m
p
;
=
sup
{
lim sup
n
→
∞
ρ
B
(
x
n
)
:
(
x
n
)
n
⊆
B
E
disjoint
}
a
m
p
;
=
inf
{
ε
>
0
:
∃
x
∗
∈
E
+
∗
so
that
B
⊆
[
−
x
∗
,
x
∗
]
+
ε
B
E
∗
}
a
m
p
;
=
sup
{
lim sup
n
→
∞
ρ
B
(
x
n
)
:
(
x
n
)
n
⊆
B
E
weakly
null
}
a
m
p
;
=
sup
{
ca
ρ
B
(
(
x
n
)
n
)
:
(
x
n
)
n
⊆
(
B
E
)
+
increasing
}
a
m
p
;
=
sup
{
lim sup
n
→
∞
‖
x
n
∗
‖
:
(
x
n
∗
)
n
⊆
Sol
(
B
)
disjoint
}
a
m
p
;
=
sup
{
lim sup
n
→
∞
sup
x
∗
∈
B
|
⟨
x
∗
,
x
n
⟩
|
:
(
x
n
)
n
⊆
B
E
disjoint
}
\begin{align*} \omega (B)&=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {disjoint} \}\\ &=\inf \{\varepsilon >0:\exists x^{*}\in E^{*}_{+} \operatorname {so}\operatorname {that} B\subseteq [-x^{*},x^{*}]+\varepsilon B_{E^{*}}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {weakly}\operatorname {null} \}\\ &=\sup \{\operatorname {ca}_{\rho _{B}}((x_{n})_{n}):(x_{n})_{n}\subseteq (B_{E})_{+} \operatorname {increasing} \}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\|x^{*}_{n}\|:(x^{*}_{n})_{n}\subseteq \operatorname {Sol}(B)\operatorname {disjoint}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\sup \limits _{x^{*}\in B}|\langle x^{*},x_{n}\rangle |:(x_{n})_{n}\subseteq B_{E}\operatorname {disjoint} \}\\ \end{align*}
for every norm bounded subset
B
B
of
E
∗
E^{*}
.