On Wikipedia there's a nice example of a non-Borel set due to Luzin. For completeness, I'll summarize it here. For $x\in[0,1]$, let
\begin{align}
x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}}
\end{align}
be the continued fraction expansion of $x$. Let $A$ be the set of numbers $x\in[0,1]$ whose corresponding sequence $a_0,a_1,a_2,\cdots$ admits an infinite subsequence $a_{k_1},a_{k_2},a_{k_3},\cdots$ such that for each $i$, $a_{k_{i+1}}$ is divisible by $a_{k_i}$. Then $A$ is not Borel measurable.
However, $A$ is an analytic set, and in particular it is Lebesgue measurable. What is the Lebesgue measure of $A$?
My thinking is that it should be zero, since the existence of such an infinite subsequence strikes me as an improbable coincidence. However, I've forgotten whatever little I knew about continued fractions, so that's far from a proof.
I tried searching this site before asking, of course. I found only one mention, here (last sentence), where $A$ is again conjectured to have measure zero.
Best Answer
I don't have a complete argument for it but I don't agree with your conjecture and believe the Lebesgue measure is 1. I might be mistaken but hope you could at least consider my position and maybe end up with a proof yourself.
Consider a much simpler system: an i.i.d sequence of integers, where each integer has a positive probability to appear. Then the probability of the Lusin event is obviously $1$. Indeed you just need one integer to be repeated infinitely often for the Lusin event to hold, but even if you insist on being a strict divisor, after you have seen an integer $k$, you are bound with probability $1$ to see $2k$ after some time, and then you could repeat the argument ensuring $4k$ will appear, etc.
Now what is the relevance of this toy model to the case at hand ? Surely the continued fraction expansion of a uniform random real number is not an i.i.d. sequence?
Well I just found these wikipedia pages:
Basically what they tell you is the following: you can easily define the continued fraction expansion of $x\in(0,1)$ with just two functions: $a_1(x) = \lfloor 1/x \rfloor$ and $h(x) = \{ 1/x\}$. You can then set $a_2 = a_1\circ h$, $a_3 = a_1 \circ h \circ h$, etc.
You can define a probability measure on $(0,1)$ by setting $\mu(dx) = \dfrac {dx}{\ln(2)(1+x)}$. Then $h : (0,1) \to (0,1)$ is a measure-preserving ergodic operator called the Gauss-Kuzmin-Wirsing operator.
Consider $X$ with distribution $\mu$. The "measure-preserving" part tells you that the sequence $X,h(X),h(h(X)), ...$ is identically distributed. The "ergodic" part tells you that the sequence, even though it's not independent, looks independent "in the long run". That is, $P(h^n(X) \in A \mid X \in B) \to P(X \in A)$ as $n$ goes to infinity.
If you now apply $a_1$ to this sequence, you end up with the continued fraction expansion of $X$, and we have seen that it is "approximately iid". Its distribution is the image of the measure $\mu$ by $a_1$, which is the above-mentioned Gauss-Kuzmin distribution, where all integers have positive probability. As a result, one could conjecture that the Lusin event has probability $1$ for $X$.
Finally, since the measure $\mu$ and the Lebesgue distribution on $(0,1)$ are mutually absolutely continuous, then what holds with probability $1$ for $\mu$ holds also with Lebesgue measure $1$.