Let $\Omega=\{A,B,C,D\}$,$\mathcal{F}=\{\emptyset,\{A,B\},\{C,D\},\Omega\}$, $P(\{A,B\})=1/2$, and $P(\{C,D\}) = 1/2$. Now $(\Omega,\mathcal{F},P)$ is a probability space.
Let $\mathcal{S}=\{1,2,3,4\}$, $\Sigma=\{\emptyset,\{1,2\},\{3,4\},\mathcal{S}\}$. Then $(\mathcal{S},\Sigma)$ is a measurable space.
Let $X$ be a function such that $X(A) = 1$, $X(B)=2$, $X(C)=3$ and $X(D)=4$. Then $X$ is clearly $(\mathcal{F},\Sigma)$-measurable, hence it is a random variable on $\Omega$. However, it seems to me that $X$ does not have a well-defined expectation. Is there something wrong in the above?
Many thanks in advance!