If $X$ is inpendent of $\mathcal{F}$, then why is for $A \in \mathcal{F}$, $\chi_{A}$ and $X$ independent

Question

As the title suggests,

let $X$ be a real-valued random variable and $\mathcal{F}$ a given $\sigma-$algebra. Let $A \in \mathcal{F}$, then it immediately follows that $\chi_{A}$ and $X$ are independent. I am struggling to see why this is the case.

My attempt:

$P(\{\chi_{A}\in C\} \cap \{X\in D\})$

Is it clear that $\chi_{A}^{-1}(C)\in \mathcal{F}$? I want to say yes, but I am not sure why.

Answer 1

Notice that $$\chi_A^{-1}(C) = \begin{cases} \Omega \quad \{0,1\} \subseteq C \\ A \quad 1 \in C, 0 \not \in C \\ A^c \quad 0 \in C, 1 \not \in C \\ \emptyset \quad \text{otherwise} \end{cases}. $$ Since $A \in \mathcal{F}$, each of these sets is in $\mathcal{F}$.