How do we arrive at the Bell polytope constructed starting from definition of behaviours?

Question

While reading Valerio Scarani's book ; Bell Nonlocality I came across section 2.4 where the author tries to represent the set $\mathcal{L}$, of all local behaviours as a polytope. The term behavior is defined (Eqn 2.4 of the book) as the set $$\mathcal{P} = \{P(a,b\ |\ x,y)\}$$ where $P$ represents the probability of getting the output pair $(a,b)$ given the input pair $(x,y)$ in a Bell's Test / CHSH-like game. And the word Local has been used for any probability distribution where $$P(a,b\ |\ x,y)=P(a|x)P(b|y)$$ holds. If I understand correctly, $\mathcal{L}$ is a set of sets, the elements of $\mathcal{L}$ are not real numbers (they are sets of real numbers), hence the set $\mathcal{L}$ lacks well-ordering within it. The rest of the section (2.4.1) of the text talks about convexity of the set $\mathcal{L}$ and claims that the as the set $\mathcal{L}$ is the convex hull of it's extremal points, which brings me back to the same confusion. What is a point within $\mathcal{L}\ ?$ I thought about considering the set $\mathcal{L}$ to contain the probabilities $P(a,b\ |\ x,y)$ directly as numbers (instead of being enclosed within sets). But that approach seems to cause more problems as two different behaviors might give different prescriptions of some given $P(a_1,b_1\ |\ x_1,y_1)$, then which prescription of this element shall we follow?

I feel like I am misunderstanding the very definition of the word "behavior" in this context. Any help is greatly appreciated.

EDIT 1 - As per Section (2.4.1) of the aforementioned book, $\mathcal{L}$ is called convex because $\forall p\in[0,1]$ and for any choice of $\mathcal{P}_{LV,1}\in\mathcal{L}$ and $\mathcal{P}_{LV,2}\in \mathcal{L}$ the convex sum $\mathcal{P}=p \mathcal{P}_{LV,1}+(1-p)\mathcal{P}_{LV,2}$ also belongs to $\mathcal{L}$. I forgot to state this definition in the originl question.

Answer 1

It's hard to give a precise answer without having read the book, but the correlations that you mention

$$ P(a,b|x,y) $$

are usually called "boxes". The set of all possible boxes is perhaps what you are after. It is the set called $Q$ in this Wikipedia article

Answer 2

If I understand correctly, $\mathcal{L}$ is a set of sets, the elements of $\mathcal{L}$ are not real numbers (they are sets of real numbers), hence the set $\mathcal{L}$ lacks well-ordering within it.

You are correct, $\mathcal{L}$ is a set of sets of non-negative real numbers in a strict mathematical sense. Of course there is no natural order in this set.

What is a point within $\mathcal{L}$ ?

As we said every point is a set of numbers. For the purposes of this discussion is better to picture it as the set of conditional probabilities given a probability measure, that is keeping $P$ fixed. Or even better just consider every point as a probability measure $P$, i.e. one possible way to assigning probabilities to events. This is not exactly equivalent but I think the details are irrelevant for the current discussion.

I feel like I am misunderstanding the very definition of the word "behavior" in this context

I think that for the purposes of this discussion is easier to think that a behavior is just a probability measure. Now given two probability measures, its convex sum is a probability measure. And the point of the discussion, as far as I remember, is that the convex sum of local behaviors, i.e. probability measures that fulfills your second equation, will be also a local behavior. That leads you to a polytope.