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...



Good examples of random variables whose image is not a measurable set?
An analytic set that is not a Borel set...see [this post](https://groups.google.com/forum/#!topic/sci.math/oEh4FzmmGxQ)\* from long ago.


Such an analytic set is a continuous image of $[0,1] \setminus \mathbb Q$, and thus a Borel image of $[0,1]$.



> 
> \*From: e...@math.ohio-state.edu (Gerald Edgar)
>  Newsgroups: sci.math
>  Subject: Re: Real Measurable, non-Borel.
>  Date: 7 Oct 1993 08:13:10 -0400
>  Organization: The Ohio State University, Dept. of Math.
>  Message-ID: <29114m$e1b@math.mps.ohio-state.edu>
>  References: 
> 
> 
> In $\lt$CEIvIr....@undergrad.math.uwaterloo.ca> emlap...@undergrad.math.uwaterloo.ca (eli lapell) wrote:
>  $\gt$What is a set of real numbers which is measurable but not Borel? Or just not 
>  Borel, period ??
> 
> 
> An explicit example of a set of real numbers that is measurable (indeed,
>  analytic) but not Borel [due to Lusin, Fundamenta Math. 10 (1927) p. 77]:
> 
> 
> 
> ```
> the set of all real numbers x with continued fraction expansion
> x = a[0] + 1/(a[1] + 1/(...))
> such that, for some positive integers r[1] < r[2] < ..., we have a[r[i]] divides 
> a[r[i+1]] for all i.
> 
> ```
> 
> Other examples of analytic sets that are not Borel can be given
>  in (complete separable) metric spaces other than the line:
> 
> 
> 
> ```
> In the space K[0,1] of nonempty compact subsets of [0,1] with the Hausdorff
> metric:  The subset consists of the uncountable compact subsets.  [Hurewicz, 1930]
> 
> ```
> 
> ${}$
> 
> 
> 
> ```
> In the space C[0,1] of real-valued continuous functions on [0,1] with the
> unform metric:  The subset consists of the differentiable functions.
> [Mazurkiewicz, 1936]
> 
> ```
> 
> 


