Let $f_n(z)$ be a sequence of holomorphic functions on some bounded domain $D$, continuously extendable to the boundary $\partial D$, which converges locally uniformly in the interior. Let us also assume that on the boundary one has pointwise convergence a.e. (of the extensions). Let us denote the limit function on $D$ as $g$, and on the boundary $\partial D$ as $h$. If for each $p \in \partial D$, the limit $\lim\limits_{z \rightarrow p} g(z)$ exists, must it be equal to $h(p)$ a.e. ? What if we instead take the case of $L^p$ convergences on the boundary with respect to some measure which is absolutely continuous with respect to the Lebesgue measure?
Update: The question concerning a.e. pointwise convergence has been answered. What remains is the case of $L^p(\partial D)$ convergence.
No, in fact $g\equiv 0, h\equiv 1$ is possible. Let $\mathbb D$ be the open unit disc. For $n=1,2,\dots$ let $K_n = \{|z|\le 1-1/n\}, C_n=\{e^{it}: 0 \le t \le 2\pi - 1/n\}.$ By Runge's theorem, for each $n$ there is a polynomial $p_n$ such that
$$|p_n| < 1/n \text { on } K_n,\,\, |p_n-1| < 1/n \text { on } C_n.$$
Then $p_n\to 0$ uniformly on compact subsets of $\mathbb D,$ and $p_n \to 1$ pointwise everywhere on $\partial \mathbb D.$