Let $X$ be a compact metric space and let $\mu $ be a Borel probability measure on $X$. The following questions regard the existence of orthonormal bases for the complex Hilbert space $L^2(X,\mu )$ with special properties.
Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ formed by continuous functions with $|f_i(x)|=1$, for every $i$ and $x$.
Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ with $|f_i(x)|=1$, for every $i$ and almost every $x$.
Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ formed by continuous functions with $\sup_{i, x}|f_i(x)|<\infty $.
Does $L^2(X,\mu )$ admit a basis $\{f_i\}_i$ with $\sup_i\|f_i\|_\infty <\infty $.
The inexistence of atoms for $\mu$ might be relevant, so you are welcome to assume this in case it helps.
If you need motivation for this question, note that for every compact abelian group $G$, with normalized Haar measure, the answer to (1) is affirmative, with the group characters forming a basis.
