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^{*} .