Abstract: Let $f$ be a holomorphic function on the unit disc, and let $(S_{n_k})$ be a subsequence of its Taylor polynomials about 0. In this talk we will see that the nontangential limit of $f$ and $\lim_{k\rightarrow \infty} S_{n_{k}}$ agree at a.e. point of the unit circle where they simultaneously exist. In the first part of this talk we will focus on this result and its applications. In the second part of the talk we will discuss a convergence theorem of harmonic measures on domains in $\mathbb{R}^{N}$ which played a key role in the proof of the above result and it is of independent interest. (Joint work with Stephen Gardiner)