May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetroric instead of semirings (like used in Halmos)? To my knowledge almost all modern measure theory or real anlysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lesbegue theory. In particular when I was learning the material, Halmos' book was very readable. Hence is my question.

Motivation: May need to explain measure theory to undergraduates next semester as I work as a probability TA. My advisor told me he believe the old theory still has much of its vitality, and everything works well. So he saw no need to introduce the new theory to (first-time) learners. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the benefit(further unification of conceptual framework, cleaned up better proofs, etc)?