I was asked to construct an ONB of $L^2[0,1]$ having functions taking at most two values. By suitably scaling, I can think them as indicator functions. So question boils down to finding countably many sets $\\{E_i\\}_{i\in\mathbb{N}}$.
I verified that corresponding indicator functions indicator functions are orthogonal if intersection of any two sets is measure $0$. Also each set must have positive measure so that indicator functions have positive norm.
I need to generate any indicator function using these sets. Then any simple function can be generated. Since simple functions are dense, we are done. This concludes that $\cup E_i$ almost covers $[0,1]$ else remaining set's indicator function cannot be generated. Combining with fact that $\lambda(E_i\cap E_j)=0$, we get $\sum\lambda(E_i)=1$.
I cannot proceed anymore. Any help would be appreciated.