Let $f_n \in W^{2,p}(\Gamma)$ ($\Gamma$ here is the $C^2-$regular boundary of a bounded, open and connected subset of $\mathbb R^3$) and suppose ${\vert \vert f_n \vert \vert}_{L^1} \le M \;\forall n$ for some positive constant $M$. I 'm trying to understand the following:
There exist $f \in \mathcal M_{+}(\Gamma)$ such that $f_n \to f$ weakly. The convergence of $\{f_n\}_n$ takes place in the weak topology of measures.
- Why does this weak convegence hold? I tried to find some weak convergence theorems in measure theory that relate somehow the $L^1-$boundedness but all I found was about probability measures which is not my case.
I would really appreciate if somebody could help me understand this implication. I had some measure theory courses in the past but we never mentioned weak convergence there.
Thanks in advance!