# Accepted nomenclature of $\sigma$-algebra components and concepts

This is a question in nomenclature. I'm looking for the correct/accepted terms-of-art to describe the following:

Let $$E=$$ { $$1, 2, 3, 4, 5$$ } be a five-element set. Let $$A=$$ { $$1, 2$$ } and $$B=$$ { $$1, 3$$ } be subsets of $$E$$. Let $$\mathcal{C_0}=$$ { $$A, B$$ } be a collection of subsets of $$E$$. Using the definitions of p-systems and d-systems (a.k.a $$\pi$$-systems and $$\lambda$$-systems, but that is not the nomenclature question), we can induce a $$\sigma$$-algebra which consists of 16 sets corresponding to the empty set and the $$2^4-1=15$$ combinatoric unions of the building-block subsets: $$\Omega=$$ { { 1 } , { 2 } , { 3 }, { 4, 5 } }.

I want to say that the smallest $$\sigma$$-algebra containing $$\mathcal{C}_0$$ discerns (or resolves) the building block-sets. It does not discern or resolve $$\{4\}$$ from $$\{5\}.$$

What are the accepted terms-of-art for discern/resolve and building-block?

A 5-element $$\Omega=E$$ induces a 32 element $$\sigma$$-algebra that contains $$\mathcal{C}_0$$, but it isn't the "smallest" $$\sigma$$-algebra. So when we say "smallest" $$\sigma$$-algebra, do we mean smallest cardinality of $$\Omega$$? This is consistent with the use of "smallest" that I've seen, but I haven't found "smallest" defined that way formally.

• I should also ask: is it acceptable to call a finite set problem a $\sigma$-algebra when it is really just an algebra, without a countability question at hand? Jan 12, 2023 at 14:24

An unambiguous answer to the question What does 'smallest' mean? is given by Cinlar in Probability and Stochastics: 'smallest' means the intersection of all $$\sigma$$-algebras that include the sets in $$\mathcal{C}_0$$.

Otherwise, as there has been no other comment, I'll assume that both "discerns" and "resolves" are proper mathematical words to describe how the above $$\sigma$$-algebra is able to tell that {1} and {4, 5} are independently resolved elements in the $$\sigma$$-algebra, but neither {4} nor {5} is discerned by the $$\sigma$$-algebra.

Lastly, I'll assume that the community has expanded the term $$\sigma$$-algebra to include finite-set algebras.