On a Theorem of Marcinkiewicz and Zygmund

The purpose of the present paper is to prove the following result: Let F(P), P= (xi, x2, • ■ • , x„), be a function harmonic for xn >0 and such that for every point Q of a set £ of positive measure on the hyperplane x„ = 0 there exists a region contained in x„>0 limited by a cone with vertex at Q and a hyperplane x„ = const, where the function is bounded. Then except for a set of measure Zero, the integral