Why "One cannot construct more than countably many independent random variables"?

"Theory of Stochastic Processes", Gusak et al., Springer, 2010:

Problem 1.10:

Prove that it is impossible to construct on the probability space $\Omega = [0, 1]$, $\mathcal{F}=\mathcal{B}([0,1])$, $\mathsf{P}=\lambda$ a family of independent identically distributed random variables $\{\xi_t, t\in[0,1]\}$ with a nondegenerate distribution.

($\lambda$ denotes one-dimensional Lebesgue measure restricted to $[0,1]$.)

Solution (also from that book):

The proof strategy is to derive a contradiction to the separability of $L^2(\Omega,\mathcal{F},\mathsf{P})$.

Assume such a family exists. Because the distribution of $\xi_t$ is nondegenerate for each $t\in[0,1]$, there exists a set $A\subset\mathcal{B}(\mathbb{R})$ such that for some (and, because of the identical distribution assumption, for each) $t\in[0,1]$ we have $\mathsf{P}(\xi_t\in A)\in(0,1)$.

For any $[0,1]\ni s\neq t$, the distance in $L^2(\Omega,\mathcal{F},\mathsf{P})$ between $\mathbf{1}\{\xi_t\in A\}$ and $\mathbf{1}\{\xi_s\in A\}$ is equal to some constant $c_A>0$; therefore, the space $L^2(\Omega,\mathcal{F},\mathsf{P})$ is not separable.

I e-mailed Professor Lovasz about this question, and here is the summary of his response:

We only want to consider collections of independent random variables with a regular conditional distribution.

This necessitates the more restrictive Ionescu-Tulcea Extension theorem, which only goes through for countably many random variables, unlike the Kolmogorov Extension Theorem, which allows one to construct uncountably many independent random variables on any measurable space with a Hausdorff topology.

As (almost) everywhere in the book we assume that the underlying probability space is standard (i.e. has a regular conditional distribution). On nonstandard probability spaces, one can construct "quasirandom" graphs, where the neighborhoods of nodes are independent events. This is a special case of the construction in Section 11.3.2.