L^2 convergence and pointwise convergence

Question

I have seen an informal note which says if $f_n \to f$ under $L^2$ norm it is pointwise convergent almost everywhere.

I know that pointwise convergence implies $L^2$ convergence under LDCT conditions but I haven’t heard anything about its converse. Is it true or not?

If it is true I think there can be some relations with Littlewood’s principles. Could someone please illuminate me about this situation? If it is true could you please prove it in the easiest way?

Thanks a lot

Answer 1

False. Let $f_n$ be listing of the characteristic functions of the intervals $[\frac {i-1} n, \frac i n)$ ($1 \leq i \leq n, n\geq 1$) in a sequence . Then $f_n \to 0$ in $L^{2}$ (w.r.t. Lebesgue measure on $(0,1)$) but the sequence does not converge at any point.

Answer 2

False. Kavi Rama gave you a counterexample.

If $f_n \to f$ in $L^2$, then there is a subsequence $(f_{n_k})$, which is pointwise convergent almost everywhere.