# What is the formal defintion of $\sigma(X_1,X_2,\ldots)$ for a sequence of random variables $X_1,X_2,\ldots$?

In probability theory one often seeks to construct the $$\sigma$$-field $$\sigma(X_{1},X_{2},\ldots)$$ for a sequence of R.V.'s $$\{X_n\}_{n\in \mathbb{N}}$$ (Assumption: $$X_n:(\Omega, \mathcal{F}) \rightarrow (\Omega',\mathcal{F'}) )$$.

I realized that I'm still not 100% sure what $$\sigma(X_1,X_{2},\ldots)$$ means. My recollection is the following definition: $$\sigma(X_1,X_{2},\ldots) = \sigma\left\{ \bigcap_{n=1}^{\infty} X_n^{-1}(A_n') \mid A_1',A_2',\ldots \in \mathcal{F}' \right\}$$. Is this correct? If not, what is the correct formal definition?

• Another convention. When $\Omega' = \mathbb R$ and no sigma-algebra $\mathcal F'$ is mentioned, we assume it is the Borel sets. This is true even if Lebesgue measure has been mentioned. The notion of Lebesgue-to-Lebesgue measurability is hardly ever useful. Nov 20, 2020 at 12:42

The definition of $$\sigma(X_1,X_2,..)$$ is it is the smallest sigma algebra which makes each $$X_i$$ measurable. It is the intersection of all sigma algebras which make each $$X_i$$ measurable. It is equal to the smallest sigma algebra which contains sets of the form $$X_1^{-1}(A_1)\cap X_2^{-1}(A_2)\cap...\cap X_n^{-1}(A_n)$$ where $$n$$ is a positive integer and $$A_i$$'s are in $$\mathcal F'$$. It is possible replace the finite intersections here by infinite intersections $$X_1^{-1}(A_1)\cap X_2^{-1}(A_2)\cap...$$ so your description is also correct.
• It is also the smallest sigma-algebra which contains sets of the form $X_n^{-1}(A)$, where $n \in \mathbb N$ and $A \in \mathcal F'$. Instead of all $\mathcal F'$, you could use only a generating set for it. Nov 20, 2020 at 12:39
• Thank you. I wonder if the following identity is also correct: $\sigma(X_1,X_2,\dotsc)= \sigma(\cup_{n=1}^{+\infty} \sigma(X_1,\dotsc,X_n))$. Is it true? Jul 7, 2023 at 11:12