Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Let $f\in L^{1}\left(\Omega\right)$ and a sequence $\left\{ f_{k}\right\} $ so that $$ f_{k}\rightarrow f $$ uniformly on each compact subset of $\Omega.$ Assume $\intop_{\Omega}f_{k}=1$ for any $k.$ Can we conclude that $$ \intop_{\Omega}f=1? $$
You may assume every function here is smooth on $\Omega$ if you want.
Thank.
No. Consider $\Omega=(0,1)$, $f_k(x)=k\cdot\mathbf 1_{(0,\frac1k)}$. Then you have $f_k\to f=0$ on each interval $[a,b]\subset(0,1)$, hence on compact subsets also. You also have $\int_\Omega f_k=1$ and $\int_\Omega f=0$.