Let our probability space
Surprisingly, yes, and the proof relies on a version of Cantor’s diagonal argument.
Proof
It is sufficient to find a sequence of independent and uniformly distributed random variables
In order to apply Cantor’s diagonal argument, we first need to write an arbitrary point
Intuitively, we need to partition the information about
We urge the reader to try finishing the proof on his or her own. Here is my approach.
Applying Cantor’s diagonal argument
So let
Let us now show that each
Remark
Although aesthetically satisfying, Cantor’s diagonal argument brought us to a conclusion which clashes deeply with our intuition. Indeed, it allowed us to effectively clone a single random variable into an infinite sequence of independent random variables. We are reminded once more that real numbers are collections of infinite information, which is something we tend to forget when we associate finite data with real numbers. And one infinite collection contains as much information as an infinity of infinite collections. The argument would have failed on probability spaces containing only rational points for example. Nevertheless, such arguments have their use in mathematics as sometimes they demonstrate a negative result - an impossibility of a certain property. In other cases like this one, they demonstrate a positive result, which cannot be observed directly, but only in terms of finite approximations.