Let $(\Omega,\mathscr{F},P)$ be a probability space. We call a real-valued random variable constant, if $ P(X=a)=1$ for some $a\in\mathbb{R}$. Show that if $\Omega$ has $n\in\mathbb{N}$ elements, then there exists at most $2^n$ independent non-constant real-valued random variables on $(\Omega,\mathscr{F},P)$.
While I could think about it in the context of discrete probability spaces, I have really no idea how to start proving it for the general case, since we are talking about non-constant and real-valued random variables here, which makes it impossible to think about it in terms of combinatorics.