What is missing is a simple description of the σ-algebra generated by ℬ.
For a mere algebra, this is easy; any ℬ can be taken as a subbase of an algebra, the symmetric unions of finite families of elements of ℬ form a base of the algebra, and the intersections of finite families of elements of the base form an algebra.
For a ring, the only difference is to use intersections only of inhabited families.
But for anything from a δ-ring to a σ-algebra, nothing like this will work.
According to the quote, which I don't quite understand,
How is an algebra of subsets generated?
In particular, how is symmetric union defined as a set operation? Union is always symmetric. I searched it on the internet, but the name seems to have a different meaning for relations.
- How is a ring of subsets generated?
Thanks and regards!