# Smallest $\sigma$-algebras generated by finite Cartesian products of sets from the coordinate $\sigma$-algebras. Which is the meaning?

Suppose we want to construct an infinite sequence of independent random variables of given distributions. Specifically, for each $$n$$ let $$X_n$$ be defined on $$\left(\Omega_n\text{, }\mathcal{A}_n\text{, }\mathbb{P}_n \right)$$ and let us set $$\Omega=\prod\limits_{n=1}^{\infty}\Omega_n\hspace{0.8cm}(\text{Countable Cartesian product})$$ and $$\mathcal{A}=\displaystyle\otimes_{n=1}^{\infty}\mathcal{A_n}$$ where $$\displaystyle\otimes_{n=1}^{\infty}\mathcal{A_n}$$ denotes the smallest $$\sigma$$-algebra on $$\Omega$$ generated by all sets of the form $$A_1\times A_2\times \ldots\times A_k\times \Omega_{k+1}\times\Omega_{k+2}\times\ldots\text{,}\hspace{0.4cm}A_i\in\mathcal{A}_i\text{;}\hspace{0.4cm}k=1,2,3,\ldots$$ That is, $$\mathcal{A}$$ is the smallest $$\sigma$$-algebra generated by finite Cartesian product of sets from the coordinate $$\sigma$$-algebras

$$\bf{Questions:}$$

1. why is $$\displaystyle\otimes_{n=1}^{\infty}\mathcal{A_n}$$ generated by finite product of sets $$A_i$$ with $$i$$ ranging from $$1$$ to $$k$$ times product of $$\Omega_i$$ with $$i$$ ranging instead from $$k+1$$ to $$\infty$$?
2. In other terms, what do they mean when stating that $$\mathcal{A}$$ is the smallest $$\sigma$$-algebra generated by finite Cartesian products of sets from coordinate $$\sigma$$-algebras? What are coordinate $$\sigma$$-algebras?

The definition is weird at first sight. Let's have a closer look.

On the product space $$\Omega:=\prod_{n =1}^\infty \Omega_n$$, we want to put a useful $$\sigma$$-algebra, call it $$\bigotimes_n \mathcal{A}_n$$. Definitely, we want to ask that the projections (= coordinate maps)

$$\pi_n: \Omega \to \Omega_n: n \geq 1$$

become measurable maps with respect to the $$\sigma$$-algebra $$\bigotimes_n \mathcal{A}_n$$. So let's just define $$\bigotimes_n \mathcal{A}_n$$ as the smallest $$\sigma$$-algebra making these projections measurable:

$$\bigotimes_n \mathcal{A}_n:= \sigma(\{\pi_n:n \geq 1\})$$

Now, what does it have to do with your question? You can check (exercise for you!) that this definition is equivalent with the one you are given. I.e. prove that

$$\bigotimes_n \mathcal{A_n}= \sigma\left(\left\{A_1\times A_2 \times \dots \times A_k \times \Omega_{k+1} \times \dots\mid A_i \in \mathcal{A_i}, i=1, \dots, k, k \geq 1\right\}\right) \quad (*)$$

(2) Means exactly what I wrote down in $$(*)$$. The coordinate $$\sigma$$-algebras are the $$\sigma$$-algebras $$\mathcal{A}_n, n \geq 1$$.