# Statement about sigma field generated by two independent random variables

Let $$X_1, X_2, X_3$$ be mutually independent random variables from $$(\Omega,\mathcal{F},P)$$ to $$(\mathbb{R},\mathcal{B})$$. Show that $$X_3$$ is independent of $$\sigma(X_1,X_2)$$, i.e., for all $$B \in \mathcal{B}$$ and $$A \in \sigma(X_1,X_2)$$, $$P(X_3^{-1}(B) \cap A) = P[X_3^{-1}(B)]P[A]$$.

My attempt:
We know from the mutual independence of $$X_1,X_2,X_3$$ that for all $$B_1,B_2,B_3 \in \mathcal{B}$$, $$P[X_1^{-1}(B_1) \cap X_2^{-1}(B_2) \cap X_3^{-1}(B_3)] = P[X_1^{-1}(B_1)]P[X_2^{-1}(B_2)]P[X_3^{-1}(B_3)]$$

Take an arbitrary $$B_3 \in \mathcal{B}$$ and $$A \in \sigma(X_1,X_2)$$. Now, $$\sigma(X_1,X_2) = \sigma(\sigma(X_1) \cup \sigma(X_2))$$.

I have worked out three cases.

First case: $$A \in \sigma(X_1) \cup \sigma(X_2)$$.
Since $$X_3$$ and $$X_1$$ (or $$X_2$$) are independent, $$P[X_3^{-1}(B_3) \cap A] = P[X_3^{-1}(B_3)]P[A]$$, and we are done.

Second case: $$A =C_1 \cap C_2$$ for some $$C_1 \in \sigma(X_1)$$ and $$C_2 \in \sigma(X_2)$$.
$$C_1=X_1^{-1}(B_1)$$ for some $$B_1 \in \mathcal{B}$$ and $$C_1=X_1^{-1}(B_2)$$ for some $$B_2 \in \mathcal{B}$$. Hence, by independence of $$X_1,X_2,X_3$$, $$P[X_3^{-1}(B_3) \cap (C_1 \cap C_2)] = P[X_3^{-1}(B_3)]Pr[C_1]Pr[C_2] = P[X_3^{-1}(B_3)]Pr[C_1 \cap C_2]$$ and we are done.

Third case: $$A =C_1 \cup C_2$$ for some $$C_1 \in \sigma(X_1)$$ and $$C_2 \in \sigma(X_2)$$.
We can show that $$A^c$$ and $$X_3^{-1}(B_3)$$ are independent by case 2. Hence, $$A$$ and $$X_3^{-1}(B_3)$$ are independent. $$\tag*{\blacksquare}$$

How do we proceed for a general set $$A$$ in $$\sigma(X_1,X_2)$$?

You can prove this using the Dynkin's $$\pi-\lambda$$ Theorem. Consider $$\{A \in \sigma (X_1,X_2): P(X_3^{-1}(B) \cap A) = P[X_3^{-1}(B)]P[A]$$. By case 2) you know that this class contains $$C_1\cap C_2$$ whenever $$C_1 \in \sigma (X_1)$$ and $$C_2 \in \sigma (X_2)$$. Now verify that this class is a $$\lambda-$$ system. Next verify that $$\sigma (X_1,X_2)$$ is the $$\sigma-$$ field generated by sets of the type $$C_\cap C_2$$ where $$C_1 \in \sigma (X_1)$$ and $$C_2 \in \sigma (X_2)$$. These sets form a $$\pi-$$system. By the $$\pi-\lambda$$ Theorem it follows that our class must contain all sets in $$\sigma (X_1,X_2)$$.
• Thank you for the answer. I couldn't prove that the set you defined (let's call it $S$) is a $\sigma$-algebra. Let $E_1, E_2 \in S$. How do we show that $E_1 \cap E_2 \in S$? Aug 21, 2022 at 9:01
• @MathewsBoban Actually, you cannot prove directly that it is a $\sigma-$algebra. I have corrected my proof by switching over to $\pi-\lambda$ Theorem: en.wikipedia.org/wiki/Dynkin_system Aug 21, 2022 at 9:25