A Description of the Sets of Lebesgue Points and Points of Summability of a Fourier Series

The set of Lebesgue points of a locally integrable function on -dimensional Euclidean space , , is an -set of full measure. In this article it is shown that every -set of full measure is the set of Lebesgue points of some measurable bounded function, and, further, that a set with these properties is the set of points of convergence and nontangential (stable) convergence of a singular integral of convolution type: for some measurable bounded function . On the basis of this result the set of points of summability of a multiple Fourier series by methods of Abel, Riesz, and Picard types is described.