Let me first state the definition of an Exhausting Sequence of Sets:
Let $X$ be a set, and let $\mathcal{S} \subset \mathcal{P}(X)$ be a collection of subsets of $X$. An exhausting sequence of sets (in $\mathcal{S}$) is a sequence $(S_n)_{n \in \mathbb{N}}$ in $\mathcal{S}$, subject to:
- for all $n$, $S_n \subset S_{n+1}$
- $\bigcup_n S_n = X$
Now I must be mistaken, because to me it seems that any $\sigma$-algebra on $X$ contains such a sequence. Take for example $S_n=X$ for all $n$. So here is my question: Is there a $\sigma$-algebra that does not have an exhausting sequence?