Pardon if this question seems trivial- I will ask only where I am stuck.
Suppose I have some probability space $\Omega = \{a,b,c,d\}$ where the $\sigma$-algebra $F$ is the collection of all subsets of $\Omega$. I'll define some random variable $Y$ such that: $$Y(a)=4, \space{} Y(b)=0, \space{}Y(c) = 4, \space{}Y(d)=2.$$
Now, I have no problem generating the different subsets- were it not for the definition of a random variable and the fact that there are two different values that yield the value $4$.
My question is two-fold:
- Is $Y$ even measurable?
- When generating $\sigma(Y)$, do I have to include both $a$ and $b$? I'm going to guess not, but my understanding is a little hazy.
Any real valued function is measurable on a space if the sigma algebra contains all subsets. So Y is measurable. $\sigma (Y)$ consists of sets of the type $Y^{-1} (A)$. To write down these sets you have see which of the values 0,2,4 is in A. The answer is $\sigma (Y)=\{ \{.\},\Omega,\{b\},\{d\},\{a,c\},\{b,d\},\{a,b,c\}, \{a,c,d\}\}$