Existence of independent events Let $(\Omega, \mathcal {F},\mathbb {P})$ be a probability space such that $\Omega$ is countable, and $\mathcal {F}=2^{\Omega}$. I want to show that it is impossible to exist a countable collection of events $A_{1},A_{2},\cdots\in \mathcal {F}$ which are independent, such that $\mathbb {P}(A_{i})=\frac {1}{2}$ for each $i$. I think showing $\mathbb {P}(\omega)\leq \frac {1}{2^n}$ for $\omega\in\Omega$ and $n\in \mathbb {N}$ might help?

Hints:
Let $A_\infty$ be the set of $\omega$ which are contained in infinitely many
$A_i$.
First, show that each $\omega\in A_\infty$
satisfies $\Bbb P(\{\omega\})=0$.
Then, use a version of the Borel-Cantelli lemma (one where independence or pairwise independence plays an important role), to show that $\Bbb P(A)=1$ holds.
Since $\Omega$ is countable, this will result in a contradiction.
