Why does $L^2$ convergence not imply almost sure convergence

What’s wrong with this argument?


Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $$\lVert f_n – f \rVert_{L^2(\Omega)} \to 0,$$ i.e.,
$$\int_\Omega(f_n – f)^2 \to 0.$$
Since the integrand is positive, this must mean that $f_n \to f$ a.e.


Why is this not true? Apparently this only true for a subsequence $f_n$ (and in all $L^p$ spaces).

Solutions Collecting From Web of "Why does $L^2$ convergence not imply almost sure convergence"

Consider the following sequence of characteristic functions $f_n \colon [0,1] \to R$ defined as follows:

$f_1 = \chi[0, 1/2]$

$f_2 = \chi[1/2, 1]$

$f_3 = \chi[0, 1/3]$

$f_4 = \chi[1/3, 2/3]$

$f_5 = \chi[2/3, 1]$

$f_6 = \chi[0, 1/4]$

$f_7 = \chi[1/4, 2/4]$

and so on.

Then $f_n \to 0$ in $L^2$, but $f_n$ does not converge pointwise.