Determine all the -measurable real-valued random variables on $Ω$.

Question

Let ($Ω$, , $P$) be a probability space,

where $Ω$ = {1, 2, 3, 4} and = ($Ω$).

Let $A$ = {1},

and consider the sub-sigma algebra of : = {∅, $A$, $A$$^c$, $Ω$}.

Determine all the -measurable real-valued random variables on $Ω$.

Answer 1

$X$ is a measurable random variable iff $\{s \in \Omega |X(s) \leq k\}$ is measurable for all $k$. Thus we would need $X(2) = X(3) = X(4)$. So all functions are of the form $X(1) = a, X(2)=X(3)=X(4) = b$.