  It is established that if a harmonic function $u$ on the unit disk $\mathbb
D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle
$\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$
also has angular limits a.e. on $E$ and both boundary functions are finite a.e.
and measurable on $E$. The result is extended to arbitrary Jordan domains with
rectifiable boundaries in terms of angular limits and of the natural parameter.
