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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.

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  • $\begingroup$ 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? $\endgroup$ Jan 12, 2023 at 14:24

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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.

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