On the existence of strong liftings in second countable topological spaces

Let X be a second countable topological space, 21 a -field o! subsets of X containing all open sets and a finite positive measure on 91, such that (X, 9l,) is a complete measure space and (U) >0 for every nonempty open U CX. Then there exists a lifting : 91-*21 which satisfies U C(U) for every open subset U CX.