  Sinai proved that a nonatomic ergodic measure-preserving system has any
Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we
show that any other Bernoulli shift that is of strictly less entropy and is
stochastically dominated by the original measure can be obtained as a monotone
factor; that is, the factor map has the property that for each point in the
domain, its image under the factor map is coordinatewise smaller than or equal
to the original point.
