= Poisson integral converges nontangentially at Lebesgue points
{c}
For $F\in L^1(\partial\mathbb D)$ and a <Lebesgue point> $\xi_0$, its <Poisson integral> tends to $F(\xi_0)$ through every fixed <Stolz region>. If $\delta=1-|z|$, the <Poisson kernel> there is at most $C_A\delta/(\delta^2+d(\xi,\xi_0)^2)$, where $d$ is shorter angular distance. Dyadic annuli about $\xi_0$ bound the near contribution by small local averages of $|F-F(\xi_0)|$ times a summable sequence $C_A2^{-k}$. The contribution outside a fixed small arc is $O(\delta)\|F-F(\xi_0)\|_1$ and vanishes. This is the local kernel argument underlying the <Fatou theorem for bounded holomorphic functions>.
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