I am studying a book (Aliprantis & Burkinshaw, "Principles of Real Analysis") that, in order to introduce the concept of measure, starts from semiring. In particular the authors state that:
"A semiring of sets is the simplest family of sets for which a measure theory can be built. It turns out that most 'reasonable' collection of sets satisfy the semiring properties" [p.94]
I have the following two questions related to the issue:
What do we gain from basing a construction of measure theory on semirings? Is there a pedagogical reason?
What are those "reasonable" collection of sets the authors most probably refer to that satisfy semiring properties, i.e. where do we actually find them elsewhere?
Clearly, I cannot really foresee all the nuances related to those questions, hence it would be really nice to actually see what happens by opting for such a construction, and why in general semirings are a good starting point/option to talk about measures.
As always, any feedback is most welcome.
Thank you for your time.