The probability-preserving form of Banach's simple Lebesgue spectrum problem asks for an invertible measure-preserving map 
on a nonatomic probability
space such that its Koopman operator on 
is unitarily equivalent to multiplication
by 
on 
, where 
is normalized Lebesgue measure
on the unit circle. Equivalently, there must be a
function 
such that
 |
is a complete orthonormal basis of 
. The requirement concerns the entire mean-zero subspace, rather than one spectral component.
