Problem :
Let $\Omega:=[0,1]$
How to define two independent random variables, which describe coin toss on the above omega?
My idea:
$\Omega=[0,1]$
$\Sigma = \mathbb{B}([0,1])$
Consider the Lebesgue measure
1-reverse coin 2-obverse coin
$X(w)= \begin{cases} 0 &\text{when } w \in [0,1/2) \\ 1 &\text{when } w\in [1/2,1) \end{cases} $
$Y(w)= \begin{cases} 1 &\text{when } w \in [0,1/4) \cup [1/2,3/4) \\ 0 &\text{when } w\in [1/4,1/2)\cup [3/4,1) \end{cases} $
It seems, that $P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$
And, for example $P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$
What do you think about this?
Yes, your idea works just fine.