  We consider the Schur-Horn problem for normal operators in von Neumann
algebras, which is the problem of characterizing the possible diagonal values
of a given normal operator based on its spectral data. For normal matrices,
this problem is well-known to be extremely difficult, and in fact, it remains
open for matrices of size greater than $3$. We show that the infinite
dimensional version of this problem is more tractable, and establish
approximate solutions for normal operators in von Neumann factors of type
I$_\infty$, II and III. A key result is an approximation theorem that can be
seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.
