  We describe majorization between selfadjoint operators in a $\sigma$-finite
II$_\infty$ factor $(\mathcal{M},\tau)$ in terms of simple spectral relations.
For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$
with trace-preserving conditional expectation $E_{\mathcal{A}}$, we
characterize the closure in the measure topology of the image through
$E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in
$\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also
obtain similar results for the contractive orbit of positive operators in
$\mathcal{M}$ and for the unitary and contractive orbits of $\tau$-integrable
operators in $\mathcal{M}$.
