We have a sample space $(\Omega, \mathcal{A}, \mathbb{P})$, where:
$$\Omega = \{1,2,3,4,5,6\}$$ $$\mathcal{A} = \{\emptyset, \{1,2\}, \{3,4\}, \{5,6\}, \{1,2,3,4\}, \{1,2,5,6\}, \{3,4,5,6\}, \Omega \}$$
$$\mathbb{P}(\{1,2\}) = \frac{1}{6} \space\space\space;\space\space\space \mathbb{P}(\{3,4\}) = \frac{1}{3} \space\space\space;\space\space\space \mathbb{P}(\{5,6\}) = \frac{1}{2}$$
The problem is to define a random variable $X:\Omega \to \mathbb{R}$ such that $\{-1, \frac{1}{2}\} \subset X(\Omega) $ and $\mathbb{E}[X] = 0$.
The first issue I have is that I'm not familiar with the notation $X(\Omega)$. I assume that's the range of the random variable? So the exercise wants me to define $X$ such that $X = -1$ for some result of the random experiment, and also $X = \frac{1}{2}$ for some other result.
I think that if we define $X(\omega) = \frac{\omega-3}{2}$, with $\omega \in \Omega$, then we could say that $X(1) = -1$ and $X(4) = \frac{1}{2}$, which means that both $-1$ and $\frac{1}{2}$ belong in $X(\Omega)$, as required.
In fact, applying $X(\omega)$ to every $\omega \in \Omega$, we get the following range:
$$X(\Omega) = \{-1,\space -\frac{1}{2},\space 0,\space \frac{1}{2},\space 1,\space \frac{3}{2}\}$$
In order to start thinking about $\mathbb{E}[X]$, we need the probability function $\mathbb{P}(X = x)$. I believe this would be $\mathbb{P}(X = X(\omega)) = p(\omega) $, where $p(\omega)$ is the probability of a given elementary event.
We should be able to get $p(\omega)$ from the probabilities of the two-element events we were given, solving the following system of equations:
$$p(1) + p(2) = \frac{1}{6}$$ $$p(3) + p(4) = \frac{1}{3}$$ $$p(5) + p(6) = \frac{1}{2}$$
Since this system has $3$ equations and $6$ unknowns, I think this is where we can state that the random variable $X$ is not unique?
At the most, we could add the following condition:
$$\mathbb{E}[X] = -1p(1) -\frac{1}{2}p(2)+0p(3)+\frac{1}{2}p(4)+1p(5)+\frac{3}{2}p(6) = 0$$
But this still would leave us with many possible probabilities to assign to the elementary events, and thus we have many possible random variables that we could define.
Does this make any sense? This is the first time I've been asked to "formally" define a random variable out of a sample space, so I'm doubting myself.
Any help is appreciated.