Before I state what my problem is I first want to give some context.
A Haar measure is a measure on the Borel subsets of a locally compact Hausdorff group $X$. The Haar measure is inner regular on open subsets, outer regular on all Borel subsets, finite on every compact subset and also left invariant. It is constructed by first defining a function $\mu$ defined only on the compact subsets of $X$. I won't go in to too much detail, but it is constructed by covering a compact subset $K$ with translations of unit neighbourhoods. Then you let the unit neighbourhood get smaller and smaller and then you can prove that there exists some limit function $\mu$. Then $\mu$ can be shown to be finitely additive, subadditive and monotone on compact subsets. We then define the function $\mu_*(U)=\sup\{\mu(K)\mid \text{$K$ is compact, $K\subset U$}\}$ defined on the open subsets of $X$, and then finally define an outer measure on all subsets, given by $\mu^*(A)=\inf\{\mu_*(U)\mid \text{$U$ is open, $A\subset U$}\}$. Then by restricting this function to the Borel subsets we finally get a measure. It can then be shown that this measure is a Haar measure.
What I want to show is that $\mu^*=\mu$ on all compact subsets of $X$. Hence let $K$ be a compact subset of $X$. I have already shown that $\mu^*(K)\ge\mu(K)$using the following proof:
For all $\epsilon>0$ there exists some open subset $U$ such that $K\subset U$ and such that $\mu^*(K)+\epsilon\ge\mu_*(U)$. Additionally $\mu_*(U)\ge\mu(K)$ hence $\mu^*(K)+\epsilon\ge\mu(K)$. This holds for all $\epsilon>0$ hence $\mu^*(K)\ge\mu(K)$.
Hence I only need to show that $\mu^*(K)\le\mu(K)$. I have tried using the following approach:
Let $U$ be an open subset. Then for all $\epsilon>0$ there exists some compact subset $C$ such that $C\subset U$ and such that $\mu_*(U)\le\mu(C)+\epsilon$. Since $\mu^*(K)\le\mu_*(U)$ it follows that $\mu^*(K)\le\mu(C)+\epsilon$. Now the problem is of course that I have to show that $\mu^*(K)\le\mu(K)+\epsilon$ and I don't know how $\mu(C)$ and $\mu(K)$ relate. I tried constructing some open subset $U$ such that $\mu(C)\le\mu(K)+\epsilon$ whenever $C\subset U$ and $K\subset U$ since then $\mu^*(K)\le\mu(K)+\epsilon$, but I couldn't manage to construct said $U$.
I really feel like I miss something very obvious and it's really nagging me. Any help would be appreciated!