Measures of weak non-compactness in 𝐿₁(𝜇)-spaces
Authors: Chen Dongyang 
Published: January 2023

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
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ω
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disjoint
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\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^{*}

.