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The principal problem considered is the determination of all nonnegative functions W(x) with period 277 such that i" \Jie)\pwie)de The main result is that W(x) is such a function if and only if 1/7-1 7(0)= iim-f fie-4>)d4> * Am.+ 77 Jes||

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. WHEEDENC) 0. A classical result of Fatou states that if u(z) is harmonic and bounded in |z| < 1 then u has a nontangential limit at almost every point eie. The same conclusion holds if u is only bounded from below. These results have a local analogue -namely, if u(z) is harmonic in |z| < 1 and at each point eie of a measurable set E there is some cone in which u(z) is bounded from either above or below then u(z) has a nontangential limit at almost every eie e E. With the aid of conformai mapping one can show the above results hold for regions of the plane more general than \z\ < I.Methods similar to those used for \z\ < 1 may be applied to functions which are harmonic in the unit ball of Euclidean (« + l)-space. However, since we lack a conformai mapping theorem for more general domains PcPn + 1 the situation there is more technically complicated. Known results for certain kinds of domains D^En + 1 are due to Brelot and Doob [2], Calderón [3], Carleson [4] and Widman [7], and the purpose of this paper is to obtain nontangential boundary values for functions which are harmonic in still more general types of domains D<=£n + 1 and bounded from above or below in cones. The domains which we consider are all regular domains for the solution of the Dirichlet problem.For a regular domain D, our result may be stated as follows. Let E^8D and suppose for each Q e E there is a cone with vertex Q which is exterior to D. Then if u is harmonic in D and is bounded from above or below in a cone with vertex Q for each Q e E, u has a finite nontangential limit at each Q e E except for a set of harmonic measure zero.It is natural that the exceptional set be one of harmonic measure zero since if pe 8D is any set of harmonic measure zero it is easy to construct a positive harmonic function with boundary value +oo at each point of P.Our main result is a consequence of the basic result that if u is harmonic and bounded in a starlike Lipschitz domain D, then u has a finite nontangential limit at each QedD except for a set of harmonic measure zero. In §1 of this paper we define the terms starlike Lipschitz domain, nontangential limit, and harmonic measure. We also include some elementary consequences of these definitions and some essential theorems on the differentiation of integrals.