Orthogonal continuous functions (Q1852319)

Let \(X\) be a compact Haussdorff space and \(\nu \) a regular Borel probability measure. The authors discuss the problem under which conditions one may find an orthogonal basis of \( L^2 (X, \nu) \) consisting of \textit{continuous} functions. The main result states that there exists a regular Borel probability measure \(\nu \) on \( X=\{ 0,1\} ^{\mathbf c} \) such that every orthogonal \textit{family} of continuous functions in \( L^2 (X, \nu) \) is at most countable but \(\nu \) is Maharam homogeneous of dimension \({\mathbf c}\) which implies that the cardinality of every orthonormal \textit{basis} in \( L^2 (X, \nu) \) has cardinality \({\mathbf c}.\) Further, the existence of an orthonormal basis in \( L^2 (X,\nu) \) consisting of continuous functions implies the existence of an orthonormal basis in \( L^2 (X,\mu) \) consisting of continuous functions provided that \(\mu \) and \(\nu \) are nice measures with \( \mu \ll \nu .\)