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So it's quite obvious that a countably infinite collection of independent Bernoulli random variables exists but as always uncountability complicates the matter.

For similar questions about uncountable collections of non-constant random variables on $[0,1]$ the answer is no, see here. But I could not find any resources talking about uncountable collections of Bernoulli random variables.

user291678
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    Bernoulli random variables are non-constant if $0<p<1$ so the statement in the linked question applies to such variables as well. But it is also limited to a particular probability space. The choice of probability space is a vital part of the statement. Are we allowed to choose a different one? – David K Oct 10 '18 at 12:00

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It is not clear whether you are interested only in random variables on $(0,1)$ or any probability space will do. Given any collection of distribution functions $\{F_i\}_{i \in I}$ we can construct independent random variables $\{X_i\}_{i \in I}$ such that $X_i$ has distribution $F_i$ for each $i$. Kolomogorov's Consistency Theorem covers this and even more general constructions.

Kavi Rama Murthy
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