Lemma 17.3.4.label Let $X$ be a set and $\alg \subset 2^{X}$, then the following are equivalent:
- (1)
$\alg$ is a $\sigma$-algebra.
- (2)
$\alg$ is a $\pi$-system and a $\lambda$-system.
Proof. $(2) \Rightarrow (1)$: Let $A, B \in \alg$, then $A \cup B = (A^{c} \cap B^{c})^{c} \in \alg$. Thus $\alg$ is an algebra. By Lemma 17.1.5, $\alg$ is a $\sigma$-algebra.$\square$