I found this on a qualifier exam, and I think it will help me understand $L^p$ spaces better.
Let $f_n$ be a sequence of measurable function on a finite measure space. Suppose that $\sup_n \int_X |f_n(x)|^2 d\mu < \infty$ and that $\lim_{n\to \infty}f_n(x) =: f(x)$ exists $\mu$-almost everywhere. Which of the following are true (proving or providing a counterexample):
(1) $\int_X |f(x)|^2 d\mu < \infty$
(2) $ \int_X |f(x)| d\mu < \infty$
(3) $\lim_{n\to\infty} \int_X |f_n(x) - f(x)|^2 d\mu = 0$
(4) $\lim_{n\to\infty} \int_X |f_n(x) - f(x)| d\mu = 0$