  In 1955, Lehto showed that, for every measurable function $\psi$ on the unit
circle $\mathbb T,$ there is function $f$ holomorphic in the unit disc $\mathbb
D,$ having $\psi$ as radial limit a.e. on $\mathbb T.$ We consider an analogous
boundary value problem, where the unit disc is replaced by a Stein domain on a
complex manifold and radial approach to a boundary point $p$ is replaced by
(asymptotically) total approach to $p.$
