(PTE) 2.1. Independence
First chapter was about the essential of measure theory. We especially focused on important result for finite measure or at least $\sigma$finite measures. We defined a probability space as a measure space and a random variable as a measurable function in it.
Following chapters will cover two fundamental theory in convergence of random variables: the strong law of large numbers (Chapter 2) and the central limit theorem (Chapter 3). We start by assumung nice but in many real cases inadequate condition  mutual independence of random variables  and modify the result to achieve the stronger one. As a starting point, this subsection covers the notion of independence. The rest of the chapter is about law of large numbers.
Independence of random variables
As I pointed out earlier, it is natural to define a property of a function as a property of the related set (its domain). We do the same here: we define independence of $\sigma$fields first.
(i) $E_1,\cdots,E_n$ are independent if $P(\bigcap\limits_{i=1}^n E_i) = \prod\limits_{i=1}^n P(E_i).$
(ii) $\mathcal{F}_1,\cdots,\mathcal{F}_n$ are independent if $P(\bigcap\limits_{i=1}^n E_i) = \prod\limits_{i=1}^n P(E_i)$ for all $E_i \in \mathcal{F}_i.$
In fact to be extra specific we say it is $P$mutually independent if the above condition is met. It gives us extra information about regarding probability measure ($P$) and that it is mutual. If we just write “independence” it means mutual independence. We drop $P$ if the measure is clear without confusion.
Independence of random variables are defined as independence of generated $\sigma$fields.
We sometimes write $X \perp Y$ for independence between $X$ and $Y$.
It is not necessary to check all possible products of the events just to check the independence of $\sigma$fields. We can use $\pi$$\lambda$ theorem.
(1) For a fixed $A \in \mathcal{A}$, let $\mathcal{L}_A = \{ B \in \mathcal{F}: P(A \cap B) = P(A)P(B) \}$ be a $\lambda$system containing $\mathcal{B}$. By $\pi$$\lambda$ theorem, $\sigma(\mathcal{B}) \subset \mathcal{L}_A$.
(2) Now for fixed $B \in \sigma(\mathcal{B})$ let $\mathcal{L}_B = \{ A \in \mathcal{F}: \mu(A \cap B) = \mu(A)\mu(B) \}$. Then similar to the above, $\mathcal{L}_B$ is a $\lambda$system that contains $\mathcal{A}$ and $\sigma(\mathcal{A}) \subset \mathcal{L}_B$ follows.
It is clear that $\mathcal{P} = \{ X^{1}(\infty, x]:~ x\in (\infty, \infty]\}$ is a $\pi$system and $\sigma(\mathcal{P}) = \sigma(X)$, so the corollary directly follows.
Existence of a sequence of independent random variables
In the following chapters, we will construct a sequence of independent random variables to state and prove limiting laws. It is important to mention that such sequence exists.
For a finite number $n\in\mathbb{N}$, we can construct $n$ independent random variables using product space. Given distribution functions $F_i$, $i=1,\cdots,n$, let $X_i$ with $P(X_i \le x) = F_i(x)$ and $X_i(\omega_1,\cdots,\omega_n)=\omega_i$. (i.e. $X_i$ is the coordinatewise projection.) Let $(\Omega, \mathcal{F}, P) = (\mathbb{R}^n, \mathcal{B}(\mathbb{R}), P)$ where $P((a_1,b_1]\times\cdots\times(a_n,b_n]) = \prod\limits_{i=1}^n (F_i(b_i)  F_i(a_i))$ then $X_i$’s are independent.
Now we need infinite number of independent random variables. Consider $\mathbb{R}^\infty := \{(x_1,x_2,\cdots):~ x_i \in \mathbb{R}\}$, an infinitedimensional product space of $\mathbb{R}$ and corresponding product $\sigma$field $\mathcal{R}^\infty$.^{1} Kolmogorov’s extension theorem states that we can construct a unique probability measure on this space.
Furthermore, if a measurable space $(S, \mathcal{S})$ is nice, that is, there is a onetoone map $\varphi: S \to \mathbb{R}$ so that $\varphi$, $\varphi^{1}$ are both measurable, then we can also construct a sequence of random elements $\{X_n\}_{n\in\mathbb{N}}: \Omega \to S.$
Acknowledgement
This post series is based on the textbook Probability: Theory and Examples, 5th edition (Durrett, 2019) and the lecture at Seoul National University, Republic of Korea (instructor: Prof. Johan Lim).

I will cover the detail later when reviewing Convergence of Probability Measures, 2nd edition (Billingsley, 1999). ↩