If limit of an integral of a positive integrand is 0, does that mean the limit of the integrand is 0? I have 
$$
\lim_{n\rightarrow\infty}\int_{\mathbb{R}} [h_n(x)-h(x)]^2dx =0
$$
I think this means
$$
\lim_{n\rightarrow\infty} h_n(x)=h(x) 
$$
for almost all $x\in {\mathbb{R}}$? Am I correct? I am wondering because people fuss over interchange of limits and integrals.

As Jonaz has pointed out, you can only guarantee convergence almost sure convergence along a subsequence.
The usual example of why this is true comes from the "typewriter" sequence of functions. Take the sequence of functions
\begin{align*}
f_0 &= \chi_{[0, 1]} \\
f_1 &= \chi_{[0, \frac{1}{2}]} \\
f_2 &= \chi_{[\frac{1}{2}, 1]} \\
f_3 &= \chi_{[0, \frac{1}{3}]} \\
f_4 &= \chi_{[\frac{1}{3}, \frac{2}{3}]} \\
f_5 &= \chi_{[\frac{2}{3}, 1]}
\end{align*}
and so on (hopefully it's clear how the sequence continues, if not please say). You can confirm that $f \to 0$ in $L^2$, but will not convergence almost surely since for any $x \in [0, 1]$, $f_n(x) = 1$ for infinitely many $n$.
