The set of Lebesgue points of a locally integrable function on $N$-dimensional Euclidean space $\mathbf R^N$, $N\geqslant1$, is an $F_{\sigma\delta}$-set of full measure. In this article it is shown that every $F_{\sigma\delta}$-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: $$ \varphi_\varepsilon\ast f(x), \quad \varphi_\varepsilon(t)=\varepsilon^{-N}\varphi(t/\varepsilon)\in L(\mathbf R^N), \quad \varepsilon\to+0, $$ for some measurable bounded function $f$. 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.