Poisson integrals and nontangential limits

A new result is established for nontangential limits of the Poisson integral of an $f\in L^{p}(\mathbf{R}^{N})$ for $N\geq 2.$ This is accomplished by showing for $N=2,\exists f$ such that the $\sigma$-set of $f$ strictly contains the Lebesgue set of $f.$ A similar theorem is also proved for Gauss-Weierstrass integrals, giving a new result for solutions of the heat equation.