  Let M be a II_1 factor, A a masa in M and E the unique conditional
expectation on A. Under some technical assumptions on the inclusion of A in M,
which hold true for any semiregular masa of a separable factor, we show that
for every discrete a in the positive part of the unit ball of A it is possible
to find a projection p in M such that E(p)=a$. We also show an example of a
diffuse operator x in A such that there exists a projection q in M with E(q)=x.
  These results show a new family of instances of a conjecture by Kadison, the
so-called "Carpenter's Theorem".
