Sigma algebra and algebra difference
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?
Solution 1:
The word 'countable' is the same as 'in bijection with the natural numbers' or 'in bijection with the integers.' There are infinitely many integers, so it's "bigger" than finite. But it's also somehow the smallest infinity.
A common case where this might come up is with respect to open and closed sets. A finite union of closed sets is closed. But an infinite union of closed sets might not be closed. For example, if we consider the sets $I_n = [\frac{1}{n}, 1 - \frac{1}{n}]$, then each $I_n$ is closed. But $\cup_{n \in \mathbb{N}} I_n = (0,1)$, an open set.
With respect to your De Morgan's law question: It is a fundamental fact that $A = B \iff A^c = B^c$, and that $(A^c)^c = A$. So they complemented your De Morgan's law to get that statement.
Finally - algebras and sigma algebras are collections of sets. To be closed under finite intersections means that taking any number of finite intersections of elements of the algebra yields an element (another set) that is in the algebra. But maybe this isn't true for an infinite intersection, etc.
Solution 2:
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.