Suppose that $X$ is a random variable that is measurable with respect to a $\sigma$-algebra $\mathcal{F}$.
Is it true that a function of $X$ must also be measurable with respect to $\mathcal{F}$?
For example, would $e^X$ be measurable here? I believe that it would be, although I am not sure how to demonstrate this.
It easily follows from the definition of a measurable map that the composition of two measurable maps is itself measurable. So any measurable map of $X$ would be measurable with respect to $\mathcal F$.
However, a non-measurable map of $X$ is not necessarily measurable.