= Fatou theorem for bounded holomorphic functions
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A bounded <holomorphic function> on the <unit disc> has finite <nontangential limits> almost everywhere on its boundary. To outline a proof, bounded real and imaginary <harmonic> parts have $L^\infty$ boundary representations: apply the <Banach-Alaoglu theorem> to their circular traces and pass to the limit in the <Poisson integral on the unit disk>. Almost every boundary point is a <Lebesgue point>. The fact that a <Poisson integral converges nontangentially at Lebesgue points> gives the desired limits. Their $L^\infty$ norm equals the interior supremum norm, and the <Poisson integral> recovers the original function. Exceptional points can lack even radial limits; $\exp(i\operatorname{Log}(1-z))$ is a bounded example at the boundary point one.
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