A holomorphic function $b$ mapping the unit disk $\disk$
into itself induces a family of measures $\tau_\alpha$,
$|\alpha|=1$, on the unit circle $\circle$ by means of
Herglotz's Theorem. This family of measures defines the
Aleksandrov operator $A_b$ by means of the formula $A_b
f(\alpha) = \int_\circle f(\zeta)\,d\tau_\alpha(\zeta)$, at
least for continuous $f$. This operator preserves the
smoothness classes determined by regular majorants, and is
seen to be compact on these classes precisely when none of the
measures $\tau_\alpha$ has an atomic part. In the process, a
duality theorem for smoothness classes is proved, improving a
result of Shields and Williams, and various theorems about
composition operators on weighted Bergman spaces are extended
to spaces arising from regular weights.