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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?).

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    $\begingroup$ There is a huge overlap of terminology in mathematics. This is normal. $\endgroup$
    – Asaf Karagila ♦
    Mar 12, 2016 at 9:16

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$\sigma$-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).

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    $\begingroup$ The fact that any $\sigma$-algebra can become a unital $\mathbb F_2$-algebra, is not that unimportant, since it gives you an easy proof, that the cardinality of any finite $\sigma$-algebra is a power of $2$. $\endgroup$
    – MooS
    Mar 12, 2016 at 9:18
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    $\begingroup$ As a note, the word "algebra" is used far more broadly than that. It's undoubtedly one of the most overloaded terms in mathematics. Other uses include universal algebra and algebras for a functor or monad. These are related to each other and to various abstract algebra notions, but don't align perfectly. For example, fields are not algebraic structures in the sense of universal algebra, but most other abstract algebra structures are. $\endgroup$
    – Derek Elkins left SE
    Mar 12, 2016 at 10:04

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