TITLE: Why "One cannot construct more than countably many independent random variables"?
QUESTION [18 upvotes]: I'm reading the book "Large Networks and Graph Limits" by László Lovász.
On the page 18 he said the following:

One cannot construct more than countably many independent random
  variables   (in a nontrivial way, neither of them   concentrated on a
  single value).

But I can not understand why it is impossible, I'm asking for your help.

REPLY [3 votes]: 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.