# Clarification regarding the definition of π- and d-systems

This is taken form A. Karr probability book.

a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then B\A ∈ D) and countable increasing union.

b) A π-system is a family of subsets closed under finite intersection.

A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.

It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.

• Usually, we are interested in $\sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $\pi$-systems and $d$-systems because we are interested in $\sigma$-algebras. – Quoka Dec 10 '18 at 5:52