Holomorphic function in unit disc?

Question

I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose $F$ is holomorphic in the unit disc, and $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ |F(re^{i\theta})|\,d\theta < \infty,$$ where $\log^+ u = \log u$ if $u\geq 1$, and $\log^+ u = 0$ if $u < 1$. Then $\lim_{r\to 1} F(re^{i\theta})$ exists for almost every $\theta$. The above condition is satisfied whenever $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi |F(re^{i\theta})|^p\,d\theta < \infty, $$ for some $p> 0$. Functions that satisfy the latter condition are said to belong to the Hardy space $H^p(\mathbb{D})$.