How to understand the operation "choose a random subset" in combinatorics?

Question

Hello I'm reading Tao and Vu's book additive combinatorics，and I can not fully convince myself to believe the proof of Theorem 1.13. In the proof, they constructed a set by the following way(It seems that I'm not allowed to upload picture now, so I type it down)

Define a set $B\subset\mathbb{Z}^+$ randomly by requiring the events $n\in B$ (for $n\in \mathbb{Z}^+$) to be jointly independent with probability $\textbf{P}(n\in B)=\min\bigg(C\sqrt{\frac{\log n}{n}},1\bigg)$, where $C$ is a large constant to be chosen later.

Since I've not seen such a method before, I have several questions:

1. If one talks about randomness, then there should be a probability space. In the proof they choose a set $B$ randomly, what is the probability space $(\Omega,\mathcal{F},\mathbb{P})$ here?
2. Relating to the first question, how could I require that events $\{n\in B\}_{n\in\mathbb{Z}^+}$ to be jointly independent?
3. Why could I require that events $\{n\in B\}_{n\in\mathbb{Z}^+}$ have the assigned probability?

Any comments should be helpful! [1]: https://i.stack.imgur.com/AWZN5.png

Answer 1

1. There may be several probability spaces. Example: when we speak about one coin toss, we may use $\Omega = \{0, 1\}$, $\mathcal{F} = 2^{\Omega}$, $P(0)=p$, $P(1) = 1-p$ with $\xi(\omega) = \omega$ or $\Omega = [0,1]$, $\mathcal{F} = \mathcal{B}[0,1]$, $P$ - standard Lebesgue measure, with $\xi(\omega) = I_{x \in [0,p]}$. Both of them are correct.

In your problem we may put $\Omega_1 = \{ (i_1, i_2, \ldots ) | i_j \in \{ 0; 1\} \}$ and $\mathcal{F} =\sigma$-algebra containing the cylindrical sets. For example, $ w = (1,0,1,1,0, \ldots )$ means that $1 \in B, 2 \notin B, 3 \in B, 4\in B, 5 \notin B, \ldots$. We also may put $\Omega_2 = [0,1]^{\mathbb{N}}$ with corresponding $\sigma$-algebra.

2-3. Put $a(n) = \min\bigg(C\sqrt{\frac{\log n}{n}},1\bigg)$ and define $P( w \in \Omega_1 : w_{i_1} = 1, w_{i_2} = 0, w_{i_3} =1) =a_{i_1} (1-a_{i_2})a_{i_3} $ and so on. It allows us to define a measure on cylindrical $\sigma$-algebra. The existence of such a measure follows from Kolmogorov existence theorem.

Remark: the explicit form of $\Omega$ is not important in such problems as it doesn't give any useful information.