Approximating by smooth functions on $W^{1,p}_\mathrm{loc}$? I'm a little bit confused about different ways of approximating by smooth functions, in particular, quasiconformal mappings.
So if a map $\phi : R\to R'$ is $K$-quasiconformal map on a relatively compact set $R$, it is true that $\phi \in W^{1,2}(R)$. And I know that $\phi$ can be approximated in $W^{1,2}(R)$ by smooth functions in $C^\infty(R)$, meaning that
$ \lim_{n->\infty}||\phi_n - \phi||_{W^{1,2}(R)} = ((|| \phi_n - \phi ||_{2})^2 + || D\phi_n - D\phi ||_{2})^2)^{1/2} = 0 $
But it seems like it's possible to find smooth functions $\phi_n$ that actually 
1) $\phi_n\to\phi$ (pointwise, or possibly uniform?)
2)$||D\phi_n -D\phi||_2 \to 0$. 
So my question is, how is this possible? (i.e. What theorem?) 
The sequence I get from the approximation theorem in $W^{k,p}$ guarantees 2), but not the first one. I think I get a.e. uniform or a.e. convergence at best...
Note: I'm not entirely sure what convergence 1) is. I'm looking at a proof given by Ahlfors that shows that q.c. maps send null sets to null sets in his "Lectures on Quasiconformal Mappings." For the purpose of the proof, the a.e. convergence is enough, but Ahlfors doesn't say "a.e.," so I'm wondering if I'm missing something...
EDIT: Here's a word-by-word quote, as requested by Willie Wong
"
THEOREM 3. Under a q.c. mapping the image area is an absolutely continuous set function. This means that null sets are mapped on null sets, and that the image area can always be represented by
$$A(E) = \int\int_E J \, dx \, dy$$
PROOF. $\phi = u+iv$ can be approximated by $C^2$ functions $u_n+iv_n$ in the sense that $u_n \to u$, $v_n \to v$ and
$\int\int |u_x - (u_n)_x|^2 \, dx \, dy \to 0$
$\int\int |v_x - (v_n)_x|^2 \, dx \, dy \to 0$ etc.
Consider rectangles $R$ such that $u$ and $v$ are absolutely continuous on all side....
"
 A: (Edited after checking the Google Books preview of the book.)
You should really check the definition that Ahlfors gave in his book. From the snipplet preview I see on Google Books, his definition of a quasiconformal maps requires that $\phi$ is a topological mapping, aka a homeomorphism of $\Omega$ onto its image. This particular implies that $\phi$ is continuous. So the standard mollifier construction will give you a sequence  $\phi_n$ that converges to $\phi$ pointwise. And on any compact subset, $\phi$ is uniformly continuous, and so $\phi_n \to \phi$ uniformly pointwise. 
In other words, the question you wanted to ask is 

If $\phi \in C^0\cap W^{1,2}$, can we find a sequence $\phi_n \in C^2$ s.t. $\phi_n$ converges to $\phi$ pointwise and $\|D\phi_n-D\phi\|_2 \to 0$. 

And the answer to that is yes. Let $\eta$ be a standard bump function and define the mollified $\phi_n = \eta_{1/n} * \phi$ via convolution in the usual way, so that $\eta_{1/n}$ is supported in the ball of radius $1/n$. Then by continuity of $\phi$, for every $x$ and for every $\epsilon > 0$, there exists a corresponding $\delta$ such that $|\phi(x) - \phi(y)| < \epsilon$ if $y$ is in the ball of radius $\delta$ around $x$. For $n > \delta^{-1}$, then, 
$$ |\phi(x) - \phi_n(x)| \leq \int \eta_{1/n}(y) |\phi(y)-\phi(x)| dy \leq \epsilon $$
using that $\int \eta_{1/n}(y) dy = 1$. It is easy to see how this proof can be modified to use uniform continuity when available. 
The convergence in $\cdot{W}^{1,2}$ you know how to prove already, so I omit. 
