A month ago, I answered a question that asked, in a nutshell, whether for a given finite set of dependent Bernoulli-distributed random variables, you could construct a set of independent Bernoulli random variables that carried the same information. I formalized the idea like this:
Let $(Ω, Σ, μ)$ be a probability space. If $X_1, X_2, …, X_n$ are Bernoulli random variables on $Ω$, then there exist independent Bernoulli random variables $Y_1, Y_2, …, Y_m$ on $Ω$ and a function $f : \{0, 1\}^m → \{0, 1\}^n$ such that for all $ω ∈ Ω$, $f(Y_1(ω), …, Y_m(ω)) = (X_1(ω), …, X_n(ω))$.
(Intuitively, the conjecture says that the values of the independent $Y_i$s suffice to determine the values of the not necessarily independent $X_i$s. Notice that $m$ need not equal $n$, and indeed may need to be much larger.)
This statement can be readily disproved by choosing a small finite probability space $(Ω, Σ, μ)$ that has no pair of independent Bernoulli random variables. But suppose we require $(Ω, Σ, μ)$ to be [0, 1] with the Borel σ-algebra and Lebesgue measure. Now we have a roomier and more familiar underlying probability space. Is the conjecture true now?