Generalizing a property of two independent random variables to more than two Notation and setup:


*

*Let all random variables be defined on a probability space $(\Omega, \mathscr A, P)$

*Let $\mathscr B=\mathscr B(\mathbb R)$ be the Borel-sigma algebra on $\mathbb R$

*A collection of random variables $(X_i)_{i\in I}$ is independent, if their generated $\sigma$-algebras $(\sigma(X_i))_{i\in I}$ are independent. 

*If $X$ and $Y$ are independent random variables, and $\mathscr C = \sigma(Y)$, then for all $C \in \mathscr C$, $X$ and $1_C$ are independent. This is because $\mathscr C = \sigma(Y) = \{Y^{-1}(B): B\in\mathscr B\},$ so that $C=Y^{-1}(B), $ for some Borel-set $B$ and $1_C = 1_{Y^{-1}(B)} = 1_B(Y)$. Then $X$ and $1_B(Y)$ are independent, because $1_B(Y)$ is a measurable function of $Y$.


Question:
I had to show the last assertion while working on a problem. Now I am wondering whether it generalizes to more than two random variables?  So my question is:

Let $X,Y$ and $Z$ be independent and $\mathscr C = \sigma(Y,Z)$. Let
  $C \in \mathscr C$. Will $X$ and $1_C$ always be independent?

I'm not sure how to tackle this, because as far as I know, $C$ cannot easily be written as a pre-image of $Y$ and $Z$, since $\mathscr C = \sigma(\sigma(Y) \cup \sigma(Z)) = \sigma(Y^{-1}(\mathscr B) \cup Z^{-1}(\mathscr B))$.
I tried writing out the sigma-algebra generated by $1_C$, it is $\{\emptyset, C, C^C, \Omega\}$. But this does not seem to lead anywhere either.
Is what I'm trying to show a known result? Or is the assertion false?
Thank you!
 A: It can be shown that: $$\sigma(Y,Z)=\{\{\langle Y,Z\rangle\in B\}\mid B\in\mathscr B(\mathbb R^2)\}\tag1$$
So if $C\in\sigma(Y,Z)$ then $C=\{\langle Y,Z\rangle\in B\}$ for some $B\in\mathscr B(\mathbb R^2)$.
Then $1_C=1_B(Y,Z)$.

In general if $f$ is a function then:$$\sigma(f^{-1}(\mathcal C))= f^{-1}(\sigma(\mathcal C))\tag2$$
For a proof of $(2)$ see this answer.
This can be applied here on the function $f:\Omega\to\mathbb R^2$ prescribed by $\omega\mapsto\langle Y(\omega),Z(\omega)\rangle$ and $\mathcal C=\mathscr B(\mathbb R^2)$, and results in $(1)$.
A: A common definition of independence of a collection of $\sigma$-algebras $\left(\mathcal A_i\right)_{i\in I}$ is that for amy toy disjoint subsets $J$ and $K$ of $I$, the $\sigma$-algebras $\sigma\left(\bigcup_{j\in J} \mathcal A_j\right)$ and $\sigma\left(\bigcup_{k\in K} \mathcal A_k\right)$ are independent. 
In your context, choose $I=\{1;2;3\}$, $\mathcal A_1=\sigma(X)$, $\mathcal A_2=\sigma(Y)$, $\mathcal A_3=\sigma(Z)$, $J=\{1\}$ and $K=\{2;3\}$.
