Prove that on $(\Omega, \mathcal{F}, P) = ([0,1], \mathcal{B}([0,1]), \lambda)$ it is impossible to construct a (continuum) family $\{X_t, t \in \mathbb{R}\}$ of independet Bernoulli random variables, i.e. $\forall t \in \mathbb{R}$ $$ \mathbb P(X_t = 0) = P(X_t = 1) = \frac 12. $$
This is an exercise from Bulinsky, Shiriaev, "Theory of stochastic processes" (on Russian).
We can match every $r \in [0,1]$ with a set $\{t \in \mathbb R \, |\, r \in \{X_t = 0\} \}$, and given there are $2^{\mathbb R}$ subsets of $\mathbb{R}$ and only $\mathbb R$ available $r \in [0,1]$ we get that most of the subsets of $\mathbb{R}$ will not match any number. I tried to extract some value from that idea, but I didn't know how to use the independency condition, except for trying to find almost identical $X_t$, which should be clearly should be here.
