Does convergence in LpL^p imply convergence almost everywhere?

If I know 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

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