We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means $ \sigma_n^{\alpha}f $ of the Fourier series of a multi-dimensional function converge to $ f $ at each Lebesgue point as $ n\to \infty $.
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