TITLE: What is meant by closed under complementation?
QUESTION [12 upvotes]: I was going through the probability and measure chapter of testing of hypothesis book by L.H. Lehman, where I found this "A class of sets that contains Z and is closed under
complementation and countable unions is a σ-field". I was not able to understand closed under complementation in this matter. Any help would be appreciated.

REPLY [11 votes]: Closed under complementation means that if a set $E\in\mathcal A$, where $\mathcal A$ is our $\sigma$-algebra, then we must have $E^c\in \mathcal A$ as well. Note that closed under countable unions and closed under complementation implies closed under countable intersection by De Morgan's laws.