After some thought, considering the recent feedback to my question, I decided to clarify a few points in an answer, since the comments do not give me enough space.
Without a countable version of the axiom of choice, it is impossible to even define the Lebesque measure. We have to work within ZFC. Now, within ZFC, we can construct Lebesque non measurable sets, the Vitali sets (and they have certain fractal characteristics).
Let’s consider a toy model. This toy model does not have a direct connection to quantum mechanics, , even if some of the terminology is borrowed from quantum mechanics.
We will first consider the double well oscillator:
y'= x - x^3
In the phase space, this system has a saddle point at (0,0) , and centers at (1,0) and (-1,0) . Each of the neutrally stable centers is surrounded by a family of small closed orbits. There are also large closed orbits that encircle all three fixed points.
In our toy model, we will say that our system is in a superposition of two states, each one associated to a stable center.
We will now consider the periodically forced double well oscillator:
y'= x - x^3 - δ·y + F ·cos(ω·t)
With δ=0.25 ,F=0.25 ,and ω=1.
For these parameter values, the system has two periodic attractors, corresponding to forced oscillations confined to the left or the right well. The basins of attraction have a complicated shape, and the boundary between them is fractal. Near the boundary, slight variation of initial conditions can lead to totally different outcomes.
In our toy model, we will call measurement, the dynamics of the periodically forced double well, for each of the two entangled systems, but we have to define entanglement.
In our toy model, entanglement corresponds to the fact that the process of measurement (the dynamics of the system represented by the periodically forced double well) starts with the same initial conditions, for each of the two entangled systems.
Under these definitions of entanglement and measurement, we see that when the first of the entangled systems ends up in the left stable well, the second system will also end up in the left stable well, and similar for the right stable well.
In order to study a phenomenon similar to the one studied by Bell (in the context of our toy model), we need to define different types of measurements, by simply varying the parameters δ, F, and ω , and the way the basins of attraction change (this is a complex problem, but it can be approached).
In the context of our toy model, we cannot follow the proof of Bell’s theorem, because of the fractal character of the domains involved in the phase space of the systems (and the connection to Vitali - type non measurable sets). That means that in our toy model, in principle, we could have correlations that go beyond classical results.
I think that something similar is happening in quantum mechanics, and we did not even touch the phenomenon of synchronized chaos yet. This toy model is only meant to prove a point of principle, and give some intuitive meaning to the issues discussed here.
Hidden variable theories can explain the strange phenomenon of quantum entanglement, without any reference to quantum non locality, which I don't think is a real phenomenon. The proof of Bell's theorem fails under certain circumstances, but nonlinearity is essential. The GHZ results can also be explained with this methodology.