If I have a random variable $X$, and a Borel measurable function $f:\mathbb{R} \rightarrow \mathbb{R}$, what would I need to show in order to prove $X$ and $f(X)$ are independent?
I know that for some Borel sets $B_1$ and $B_2$, proving $P(X\in B_1, f(X)\in B_2)=P(X\in B_1)P(f(X)\in B_2)$ would show independence. However, clearly $X$ and $f(X)$ can't be independent...unless $X$ is independent of itself, right? Am I thinking about this correctly?
Note that $X$ and $f(X)$ are independent implies $f(X)$ is independent of itself, and that $f(X)$ is independent of itself iff $f(X)$ is almost surely constant.
From this you cannot conclude that $X$ is almost surely constant if no further information about $f$ is given.
Example: let $X$ take values $-1$ and $1$ with probability $\frac 1 2 $ each and $f(x)=x^{2}$. Then $f(X)=1$ and this makes $X$ and $f(X)$ independent. But $X$ is not a constant.