TITLE: Example of an algebra which is not a σ-algebra.
QUESTION [30 upvotes]: I have troubles with constructing an example of an algebra of sets which is not a σ-algebra. Could you please help me with this?

REPLY [14 votes]: Let $L$ be the collection of all finite disjoint unions of all intervals of the form:
$(−\infty, a], (a, b], (b, \infty), \emptyset, \mathbf{R}$.
Then $L$ is an algebra over $\mathbf{R}$, but not a σ-algebra because
union of sets $\left\{(0,\frac{i-1}{i}]\right\}$ for all $i \ge 1 = (0, 1) \notin L $.