Let me introduce the definitions in the title first: Standard probability space means $L^2([0,1],\mathcal B([0,1]),\lambda)$, where $\lambda$ means the Lebesgue measure. The term "independent" in title is in the sense of probability, i.e. $\mathbb P(AB)=\mathbb{P}(A)\mathbb P (B)$.
Background: Recently I am learning measure theory and I know there exists lots of bases in such space (trigonometric basis, Haar basis and so on). But I am curious about does there exist an independent basis?
Since, take Haar basis $\{Z_n\}_{n\ge1}$ as an example, we can check that for any $n\not=m$, $\lambda(\{Z_n=0\}\{Z_m=0\})\not=\lambda(\{Z_n=0\})\lambda(\{Z_m=0\})$. As to trigonometric basis $\{1,\sqrt 2\sin(2\pi nx),\sqrt2\cos(2\pi nx)\}_{n\ge1}$, there exists a deterministic relationship $\sin^2(2\pi nx)+\cos^2(2\pi nx)=1$, so they are not independent in the probabilistic sense as well.
From this point of view, I believe the answer to this question is no, since somehow I have a feeling that from information sence the independent basis contains inadequate information to represent all the random variables in such a standard probability space. But I don't know how to prove it. Any valuable advice is appreciated.
