TITLE: Does convergence in $L^p$ imply convergence almost everywhere?
QUESTION [107 upvotes]: If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?

REPLY [23 votes]: Consider the "typewriter sequence" defined by the formula:
$$f_n:=1_{\left[\frac{n-2^k}{2^k},\frac{n-2^k+1}{2^k}\right]}$$
where $k$ is an integer such that $2^k\leq n<2^{k+1}$, the sequence converges to zero in $L^p$ norm, but not pointwise.
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