Conditions on $f$ for $f(X_1),\dots,f(X_n)$ to be iid

Question

Let $X_1,\dots,X_n$ be iid non negative random variables. What is are necessary and sufficient conditions of a function $f$ for the random variables $f(X_1),\dots,f(X_n)$ to be iid?

For example is it true if $f$ is non negative and continuous?

Answer 1

It is sufficient that $f$ is measurable. That is the inverse of an open subset of $\mathbb{R}$ is a measurable subset of $\mathbb{R}$.

In particular, if $f$ is continuous it is measurable so it will definitely work for continuous functions $f$.