Let $X$ be an $\mathbb{R}$-valued random variable and $F_X$ be its cumulative distribution function. I am interested in the connection between the $\sigma$-algebra generated by $X$ and that generated by $F_X (X).$ It is obvious that if $F_X$ is strictly increasing, then these coincide. If not (i.e. $F_X$ "stays" on some interval), then just observing $F_X(X)$ does not allow you to recover $X,$ so $\sigma (F_X (X))$ cannot be greater than $\sigma (X).$ I am wondering if there is an example where $\sigma (X)$ is strictly greater than $\sigma (F_X (X)).$
Thanks!
$F_X$ is always right-continuous, so if $F_X$ is not strictly increasing it is some constant $c$ on an interval $[a,b)$. This can only happen if $\mathbb{P}(X=a) > 0$ and $\mathbb{P}(X \in (a,b)) = 0$, so observing $F_X(X) = c$ implies $X=a$. Therefore $\sigma(F_X(X)) = \sigma(X)$.