Correlation of boundary behavior of conjugate harmonic functions

It is established that if a harmonic function $u$ on the unit disk $\mathbb D$in$\mathbb C$hasangularlimitsonameasurableset$E$oftheunitcircle$\partial\mathbb D$,thenitsconjugateharmonicfunction$v$in$\mathbb D$alsohasangularlimitsa\mathrm{.}e\mathrm{.}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.