Lambda-system

Definition

A $\lambda$-system $\mathcal{D}$ is a Family of Sets satisfying the following properties

• $\Omega \in \mathcal{D}\text{or}\mathcal{D}\ne \varnothing$
• $\forall A,(A\in \mathcal{D}\Rightarrow A^{\complement }\in \mathcal{D})\text{or}\forall A,B\in \mathcal{D},(A\subset B\Rightarrow B-A\in \mathcal{D})$
• $\forall \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{D},\underset{i=1}{\overset{\infty }{⨄}}{A}_i\in \mathcal{D}\text{or}\forall \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{D},(A_1\subset A_2\subset \cdots \subset A_{\infty }\Rightarrow \bigcup _{i=1}^{\infty }{A}_i\in \mathcal{D})$

where $⨄$ is a union of the pairwise disjoint sets, and $\Omega$ is a universal set