# Relation between an "algebra" in Analysis and in Algebra

While studying some measure theory, I came across the following two definitions, given a set $$\mathbb{X}$$:

1. A ring is a non-empty subset of $$\mathcal{P}(\mathbb{X})$$ such that it is closed for unions and set-differences
2. An algebra is a non-empty subset of $$\mathcal{P}(\mathbb{X})$$ such that it is a ring and contains $$\mathbb{X}$$; equivalently, it is closed for unions and complements

This immediately got me thinking about the algebraic implications. Given a former question of mine on this site, I saw that the first definition actually said such a set was a subring of $$\mathcal{P}(\mathbb{X})$$, with the operations $$A \Delta B = (A\setminus B) \cup (B \setminus A)$$ and $$A\cup B$$.

But this left me with the second one: algebras, as far as I've seen, are defined over fields. So what field is that algebra talking about? If there isn't any, why is the term "algebra" used?

As a side note, is there an Algebraic equivalent of "$$\sigma$$-algebras" and "$$\sigma$$-rings", without talking about the particular context of subrings of $$\mathcal{P}(\mathbb{X})$$?

• (finished what I had been working on) Regarding your side note, see Representation of sigma complete Boolean algebras and Complete Boolean algebra not isomorphic to a $\sigma$-algebra. As for "algebra", the basic relations involving unions and intersections and such of sets had been, in the early 1900s (and perhaps earlier), referred to as algebraic operations on sets, so at some point instead of fields of sets (term still in use), I suppose "algebras of sets" came be be used. Commented Aug 17, 2021 at 21:12
• See also Were $\sigma$-algebras defined for probability?, including the comments. I don't know whether the use of these terms for collections of sets can be found earlier than 1927, but at least for now 1927 is an upper bound for this usage of these terms, including "ring", which is probably the most significant term here (since as I've said and you already know, "algebra" was long before this used as an adjective for many mathematical descriptions). Commented Aug 18, 2021 at 7:42