I've been reading Schilling's Measures, Integrals, and Martingales, and I ran across a remark (on page 21) that I don't understand.
Here is the setup: let $A_{1},\ldots,A_{n}$ be non-empty, disjoint subsets of X with $\cup A_{n}=X$. Schilling says that "a set A in a $\sigma$-algebra $\mathscr{A}$ is called an atom, if there is no proper subset $\emptyset\neq B \subsetneq A $ such that $B \in \mathscr{A}$. In this sense all $A_{n}$ are atoms [of the $\sigma$-algebra generated by $A_{1},\ldots,A_{n}]$."
Why are the $A_{n}$ atoms? This seems intuitive, but I don't see how to prove it.