May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of $\sigma$-rings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lebesgue 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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)

May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of $\sigma$-rings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lebesgue 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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)

May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of semirings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lebesgue 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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)

May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of $\sigma$-rings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lebesgue 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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)

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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)

May I ask what is the (historical) reason we adopted the $\sigma$-algebra rhetoric instead of semirings (like used in Halmos)? To my knowledge almost all modern measure theory or real analysis textbooks use $\sigma$-algebra nowadays, but it seems the old theory is perfectly legitimate to teach students basics of Lebesgue 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, though the class itself does not involve measure theory. 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. In fact, he is going to publish a new book in GTM series using the old theory. While this is certainly true, I do not understand what pushed the historical conceptual change. What is the real benefit(further unification of conceptual framework, cleaned up better proofs, generalized easily for non-archemedian fields, etc)?

(For an example of my past unsuccessful effort to explain measure to undergraduates, see here)