Does convergence in LpL^p imply convergence almost everywhere?

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 $x$ ?

Answer

Let $\rho(x)=\chi_{[0,1]}$ be the characteristic function of the interval $[0,1]$ . Then take the “dancing” sequence

$f_n(x) = \rho(2^mx-k)$

where $n=2^m+k$ with $0\leq k<2^m$ . This sequence converges to $0$ in $L^p$ but for any $x\in(0,1)$ we have $f_n(x)$ is not convergent.

However, it is a general fact that one can always extract a subsequence converging almost everywhere to $f$ .

Attribution
Source : Link , Question Author : 187239 , Answer Author : timur