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  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nFor $\\theta \\in \\R$ and $\\alpha \u003e 1$, let $\\Gamma_\\alpha(\\theta) = \\set{z \\in \\mathbb{D} : \\abs{z - e^{i\\theta}} \u003c \\alpha (1 - \\abs{z})}$ be the Stolz angle at $e^{i\\theta}$.\nSay that a function $u$ on $\\mathbb{D}$ has nontangential limit $L$ at $e^{i\\theta}$ if $u(z) \\to L$ as $z \\to e^{i\\theta}$ inside $\\Gamma_\\alpha(\\theta)$, for every $\\alpha \u003e 1$.\n\n1. Let $\\varphi$ be a Lebesgue integrable function on the circle ([[Lp space|SASP14MA1Z]] with $p = 1$), and let $u$ be its [[Poisson integral|TCYSWCTYYM]],\n$$u(r e^{i\\theta}) = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\frac{1 - r^2}{1 - 2r \\cos(\\theta - t) + r^2} \\, \\varphi(e^{it}) \\, dt.$$\nThen for [[almost every|RARTMA0ZEC]] $\\theta$, $u$ has nontangential limit $\\varphi(e^{i\\theta})$ at $e^{i\\theta}$.\n2. Let $f$ be a bounded [[holomorphic function|0DG0Z72CQD]] on $\\mathbb{D}$.\nThen for almost every $\\theta$, $f$ has a nontangential limit at $e^{i\\theta}$.\nIn particular the radial limit $\\lim_{r \\to 1^-} f(r e^{i\\theta})$ exists for almost every $\\theta$.\n\n## Remarks\n\nProof idea: the nontangential maximal function of $u$ is bounded by a constant times the Hardy–Littlewood maximal function of $\\varphi$, which is finite almost everywhere.\nThe limit holds for continuous $\\varphi$, and the maximal estimate extends it to all integrable $\\varphi$.\nFor part 2, the functions $f(r_k e^{it})$ have a weak-star limit $\\varphi$ in $L^\\infty$ along some sequence $r_k \\to 1$, and $f$ is the Poisson integral of $\\varphi$, so part 1 applies.\n\nExample: the singular inner function $\\exp\\left(-\\frac{1 + z}{1 - z}\\right)$ is bounded and has radial limit of modulus $1$ at every point $e^{it} \\ne 1$, while its radial limit at $1$ is $0$.\n\nBoundedness cannot simply be dropped: there are holomorphic functions on $\\mathbb{D}$ with a finite radial limit at almost no point.\nUsing the [[inner-outer factorization|X6ZQPAQF76]], part 2 extends to every function in the [[Hardy space|PNNNFJVF6W]] $H^p$, $0 \u003c p \\le \\infty$.\n\n## Explanations\n\n### L1 human\n\nA bounded holomorphic function on the disc may oscillate near the boundary, but not too wildly.\nFatou's theorem says that, approaching almost any point of the circle, the function settles down to a limit.\nSo the function has well defined boundary values, up to a set of length zero.\nThe same is true for the harmonic extension of any integrable function on the circle, and the limit is the function you started with.\n\n### L2 human\n\nFatou's theorem is the reason Hardy spaces can be viewed as subspaces of $L^p$ of the circle.\nIt is used to define [[inner functions|NN8XH5RJYB]] and [[outer functions|KGJD4A4ATY]] through their boundary values.\nThe approach must be nontangential: Littlewood showed that limits along curves tangent to the circle can fail at almost every point.\nThe proof is a model case of the maximal function method in harmonic analysis.\n",
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