Let $f_n(x).dx$ be measures with density wrt Lebesgue measure, $f_n$ derivable, $f_n \rightarrow f$ in uniform norm; $\ n \rightarrow \infty$.
Assume that they are uniformly limited in variation, i.e. $\exists M:\ \int |Df_n|(dx)<M\ \forall n$, where $Df$ is the distributional derivative of $f$.
This means $\forall \phi \in {C_c}^1(\mathbb{R}^n),\ \int f(x)\frac{\delta \phi}{\delta x_i} (x)dx=\int\phi(x)(D_i f)(dx)$.
Than there is a subsuccession $f_{k_n}\ n \in \mathbb{N}$ such that $f_{k_n}(x).dx$ converges weakly. How can i show it?
Original De Giorgi paper says it's an "easy" consequence of a theorem of de La Vallee Poussin.
Notice that i have very few confidence with this field, so please try to be as detailed as possible.