I want to define a probability space $([0,1],\mathcal F,\mathbb P)$ where $\mathbb P([0,x])=x$.
I would like to prove that there is no probability measure on $2^{[0,1]}$, and that there is a unique probability on $[0,1]$ s.t. $\mathbb P([0,x])=x$, and for such a space $\mathcal F=\sigma (\{[0,x]\mid x\in [0,1])$. My teacher said that it's a hard exercise, but do you have a reference for such a proof ?