Suppose $\Omega = \{1,2,3,4,5,6\}$, and let $\mathcal F= \sigma(\{1,2,3,4\},\{3,4,5,6\})$. Determine whether
$$X(\omega)=\begin{cases}
2 & \text{ if } w=1,2,3,4\\
7 & \text{ if } w=5,6\
\end{cases}$$
is a random variable over $(\Omega,\mathcal{F})$ and provide an example of a function on $\Omega$ that is not a random variable over $(\Omega,\mathcal{F})$.
My try:
$(i)$ We have that
$$\begin{align}
\mathcal{F} &= \sigma(\{1,2,3,4\},\{3,4,5,6\})\\\\
&= \{\emptyset,\Omega,\{1,2,3,4\},\{3,4,5,6\},
\{5,6\},\{1,2\},\{1,2,5,6\}, \{3,4\}\}
\end{align}$$
For $X(\omega)$ to be a random variable, we need $X^{−1}(B)\in\mathcal{F}$. We have
$$X^{-1}(\omega)=
\begin{cases}
\emptyset & \text{ if } 7\notin B, 2 \notin B\\
\Omega & \text{ if } 7\in B, 2 \in B\\
\{1,2,3,4\} & \text{ if } 7\notin B, 2 \in B\\
\{5,6\} & \text{ if } 7\in B, 2 \notin B\\
\end{cases}\quad\in\mathcal{F}
$$
so $X(\omega)$ is a random variable. (I wasn't sure if I need to account for the other possible subsets $A\in\mathcal{F}$ in the inverse image mapping. For example would it not be the case that $X^{-1}(\{2\})=\{3,4\}$ as well?
$(ii)$ Consider
$$X(\omega)=\begin{cases}
2 & \text{ if } w\in\{1,2,3\}\\
7 & \text{ if } w\in\{4,5,6\}
\end{cases}$$
This is not a random variable over $(\Omega, \mathcal{F})$ since
$$X^{-1}(\{2\})=\{1,2,3\}\notin\mathcal{F}$$
Is my reasoning correct?
Best Answer
Yes, your reasoning is correct. I would like to add a remark : When you want to prove that some function $X : (\Omega, \mathcal{F}) \rightarrow (E, \mathcal{G})$ is a random variable, you do not need to check that $X^{(-1)}(B)$ is an element of the $\sigma$-algebra $\mathcal{F}$ for all sets $B \in \mathcal{G}$. It is enough to check that it is the case for a family of sets $B$ that generate the $\sigma$-algebra $\mathcal{G}$.
In your case, the $\sigma$-algebra $\mathcal{G}$ is the collection of Lebesgue-measurable sets, or the discrete $\sigma$-algebra on $\lbrace 2, 7 \rbrace$. If you consider the collection of Lebesgue-measurable sets, you cannot really avoid dealing with the four cases. However, if you consider the discrete $\sigma$-algebra on $\lbrace 2, 7 \rbrace$, it is enough to study the sets $X^{(-1)}(\lbrace 2 \rbrace)$ and $X^{(-1)}(\lbrace 7 \rbrace)$.