TITLE: Sigma algebra and algebra difference
QUESTION [26 upvotes]: An algebra is a collection of subsets closed under finite unions and intersections.
A sigma algebra is a collection closed under countable unions and intersections.

Whats the difference between finite and countable unions and intersections? Does "countable" mean it implies there can be infinitely many unions and intersections?

Secondly, I was reading a definition
For an algebra on a set: By De Morgan's law, $A \cap B = (A^c \cup B^c)^c$, thus an algebra is a collection of subsets closed under finite unions and intersections.
What law are they using here to get $A \cap B = (A^c \cup B^c)^c$? I thought de morgan's law was $(A\cap B)^c = A^c \cup B^c$?
Finally, what exactly do they mean by "closed under finite unions and intersections?

REPLY [8 votes]: Ok, am not a mathematician or a student - BUT Doob - measure theory
had a pretty terrific answer for it.
I was actually looking for the answer and got in here, and obviously could not understand the answer.
So, according to Doob, here is the answer:-
The algebra S is $\sigma$ algebra if, S contains the limits of every monotone sequence in S. Notice that this is the same idea - we use for complete metric space.
So, now I believe I get it. There is a one to one similarity between this measure stuff and metric stuff, and what is a complete space there, becomes sigma algebra here.
Hope I do understand it.