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.

  • $\begingroup$ 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. $\endgroup$ – Quoka Dec 10 '18 at 5:52

Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.


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