Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?

I am teaching Doob's lemma (for two real-valued random variables $X$ and $Y$, $X$ is $\sigma(Y)$-measurable iff there exists a Borel-measurable function $f:\mathbb{R}\to\mathbb{R}$ such that $X=f(Y)$) and the main difficulty in the proof comes from the fact that $Y(\Omega)$ is in general not a Borel set. So I am wondering if there is a "natural" example that I can use to convince 4th year students that this "pathology" can naturally come up.

It is easy to construct examples, e.g., choose $A\subseteq \mathbb{R}$ any set that is not a Borel set, and equip it with the $\sigma$-algebra $\mathcal{A}=\{A\cap B; B\in \mathcal{B}(\mathbb{R})\}$, where $\mathcal{B}(\mathbb{R})$ denotes the $\sigma$-algebra of Borel sets in $\mathbb{R}$. Then the inclusion $X:(A,\mathcal{A})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable and has $A$ as image, so its image is not a Borel set. But this feels like cheating...