Show that if $\Omega$ has $n \in \Bbb N$ elements, then it exist at most $2^n$ independent non-constant real-valued random variables on $(\Omega,\mathcal F, P)$.
A real-valued random variable is called constant if $P(X=a)=1$ for some $a \in \Bbb R$.
The only idea that I have in mind for now is to somehow use variance since a random variable in not a constant one if its variance is not $0$. Also I suppose that the given sigma-algebra has exactly $2^n$ elements so it might be applied in the solution ?
I just really a starting point for this proof. Any hint will be highly appreciated.
If $X$ is a nonconstant random variable, then there exists a set $A_X \in \sigma(X)$ with $P(A_X) \notin \{0, 1\}$. If $Y$ is independent of $X$, then $A_X \notin \sigma(Y)$, since the independence would imply $P(A_X) = P(A_X \cap A_X) = P(A_X)^2$. But this can't be, since $P(A_X)$ is neither $0$ nor $1$.
Now let $M = \{X_1, \ldots, X_m\}$ be a set of independent nondegenerate random variables. Then the map $X \mapsto A_X$ is an injection from $M$ to the power set of $\Omega$, which means $|M| \le |\mathcal{P}(\Omega)| = 2^n$.