This is the definition of "measurability":
Let $\mathit X$ be a random variable defined on $(\Omega,\mathcal F,\mathbb P)$. Let $\mathcal G$ be a $\sigma$-algebra of subsets of $\Omega$. If $\sigma(\mathit X) \subseteq \mathcal G$, we say that $\mathit X$ is $\mathcal G$-measurable.
I can understand this definition, but what does it mean that "a random variable $\mathit X$ is $\mathcal G$-measurable if and only if the information in $\mathcal G$ is sufficient to determine the value of $\mathit X$"?
My questions are:
- What is the "information" in $\mathcal G$? As far as I know, $\mathcal G$ is just a set of subsets of $\Omega$. Thus $\mathcal G$ contains lots of "events". Then, when we say "information", what exactly do we mean？ I can't imagine other information except for the elements of $\mathcal G$.
- How can the "information" in $\mathcal G$ determine the value of $\mathit X$? Given $\mathcal G$, what we know is just "the elements of $\mathcal G$". How could this "infromation" help us determine the value of $\mathit X$?
Example:
This is an example of rolling a die with $\Omega = \{1, 2, 3, 4, 5, 6\}$:
$$\mathit X_1(\omega)=\omega$$
$$\mathit X_2(\omega)=\begin{cases} 1, & \omega\in \{1,3,5\} \\-1, & \omega\in \{2,4,6\}\end{cases}$$
$\mathit X_1$ and $\mathit X_2$ are both random variables. $\mathit X_1$ gives the exact outcome of the roll, and $\mathit X_2$ is a binary variable whose value depends on whether the roll is odd or even.
Let $\mathcal G = \{\emptyset, \Omega, \{1,3,5\}, \{2,4,6\}\}$, then $\mathit X_2$ is measurable w.r.t $\mathcal G$ but $\mathit X_1$ is not measurable w.r.t $\mathcal G$. I know this because I can check that $\sigma(X_2)=\mathcal G$ according to the defination of $\sigma(X_2)$.
Here are the questions:
What's the information in $\{\emptyset, \Omega, \{1,3,5\}, \{2,4,6\}\}$? What else can we know except for those four elements in $\{\emptyset, \Omega, \{1,3,5\}, \{2,4,6\}\}$?
If this is the only 'information' that we can obtain from $\mathcal G$ (i.e., there are four elements in $\mathcal G$, and those elements are $\emptyset$, $\Omega$, $\{1,3,5\}$ and $\{2,4,6\}$), how can we determine what the value of $\mathit X_2$ will be?