Help with some unfamiliar set notation

[E'lir Kramer]

I am reading http://walrandpc.eecs.berkeley.edu/126notes.pdf on the theory of random processes. The authors are making use of some unfamiliar notation early on, and I don't want to move on without understanding their formalisms.

First one is the := operator and a union operator that looks like it's being used as a variable.

From the text, section 2.3:

"We want to be able to say that if the events $A_{n}$ for $n = 1,2,...,$ are such that $A_{n} \subset A_{n+1}$ for all $n$ and if $A:=\bigcup _{n}A_{n}$, then $P(A_{n}) \uparrow P(A)$ as $n \rightarrow \infty$."

What does $A:=\bigcup _{n}A_{n}$ mean? How do you pronounce that phrase in English?

What does the up arrow mean in the next phrase? Is this a strange way of expressing "the limit of $P(A_{n})$ as n goes to infinity is $P(A)$? (Which makes sense in context).

 

[Marioeden]

I think (and I may be wrong) that if they're using standard convention, then := means "define".

In other words, that statement means "Define A as the union of the sets A(n) over all n"

And I believe that the up arrow indeed means the limit.

 

[E'lir Kramer]

Oh, I think I get it. It is fair to say that the big cup in $\bigcup _{n}A_{n}$ should read like $\sum_{n}A_{n}$ in big sigma notation?

 

[LCKurtz]

I think (and I may be wrong) that if they're using standard convention, then := means "define".

In other words, that statement means "Define A as the union of the sets A(n) over all n"

And I believe that the up arrow indeed means the limit.

I think the up arrow means the sequence $P(A_n)$ increases to $P(A)$.

 

[Marioeden]

Oh, I think I get it. It is fair to say that the big cup in $\bigcup _{n}A_{n}$ should read like $\sum_{n}A_{n}$ in big sigma notation?

Exactly, the only difference is the operation i.e. sigma means to sum and U means to take the union. So Sigma over n means summing over all n in the given range, and U over n means taking the union of the sets over the given range.

 

[E'lir Kramer]

Here's another.

A probability space is a triplet {$\Omega$, F, P} where
$\Omega$ is a nonempty set, called the sample space
F is a collection of subsets of $\Omega$ closed under countable set operations, called a $\sigma$-field. The elements of F are events.
P is a countably additive function from F into [0,1] such that P($\Omega$) = 1, called a probability measure.

...

Example 2.7.2 Pick three balls without replacement from an urn with fifteen balls that are identical except that ten are red and five are blue. Specify the probability space.

One possibility is to specify the color of the three balls in the order they are picked. Then

$\Omega = {R,B}^{3}, F = 2^{\Omega}$...

Now, what does $2^{\Omega}$ mean in this context? Omega is a set; how do you raise an integer by a set?

 

[E'lir Kramer]

Exactly, the only difference is the operation i.e. sigma means to sum and U means to take the union. So Sigma over n means summing over all n in the given range, and U over n means taking the union of the sets over the given range.

Makes sense, thanks.

 

[tiny-tim]

Hi E'lir Kramer!

Now, what does $2^{\Omega}$ mean in this context? Omega is a set; how do you raise an integer by a set?

2^Ω is the power set of Ω, ie the set of all subsets of Ω

see http://en.wikipedia.org/wiki/Power_set#Representing_subsets_as_functions

 

[E'lir Kramer]

Thanks for the link, Tim! That fully clears it up for me.

One of the biggest frustrations with mathematics self-study is that it math isn't a linear progression of knowledge, but rather a web of knowledge, and whatever thread I pursue is constantly touching other threads which are still in the dark for me. I really wish I had taken up this interest in college, where there are teachers to smooth the way. However whenever the kind teachers on this board step in and guide me, I know that I'll be able to do this on my own.

This is just a bit of set theory notation that I hadn't encountered before, but the concept of "set of all subsets" is easy enough to grasp, and so is the motivation for the notation. So this tangent won't be too derailing. Thanks again.