Algebraic structure of σ-algebras I'm confused about what algebraic structure σ-algebras belong to. Are they really algebras over fields (i.e. vector spaces with a bilinear product)? If so, what is the base field?
To me, σ-algebras seem more like lattices, because of the way intersection and union behave (they follow the absorption law, right?).
 A: $\sigma$-algebras are special cases of Boolean algebras.
They have nothing in particular to do with algebras over a field -- except in the sense that any Boolean algebra can become an unital algebra over $\mathbb F_2$ by declaring intersection to be the multiplication, symmetric difference to be the addition, and mapping $0$ and $1$ of $\mathbb F_2$ to $0$ and $1$ of the Boolean algebra. (In the case of a $\sigma$-algebra or other subset algebra, $0$ becomes the empty set and $1$ becomes the entire base set).
In general, the term "an algebra" is used in a number of not quite compatible senses about algebraic structures that have an "addition" and a "multiplication" that satisfy a distributive law. The addition is usually, but not always, assumed to have an abelian group structure, but the multiplication can more often be "strange" -- consider for example Lie algebras where it is not associative. In this fuzzy sense, Boolean algebras qualify as "algebras" because intersection (conjunction) distributes over union (disjunction).
