Let $G$ be a abelian, locally compact group with a Haar measure $dx$ on $G$. We know that every Haar measure is inner regular, ie... for any open subset $U$ of $G$, then $dx(U)=\sup\{dx(F):\;F\;\;\text{is compact and is a subset of}\;\;U\}$.Here we denote $dx(F)$ is the measure of $F$ respect to $dx$.
Now we are given a non-negative function $u:G\to\mathbb R$, an open subset $U$ of $G$ such that $u\in L^1_{loc}(G)$. We assume that, there is a constant $c>0$ so that $\int_F u(y)dy\leq c<\infty$ for any $F$ is compact and is a subset of $U$.
Is it true that $\int\limits_Uu(y)dy=\sup\limits_F\;\int\limits_F u(y)dy$, where supremum is taken over all compact $F$ which is a subset in $U$? Could you give me some ideas to prove if it is true.