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.... "