Before I state what my problem is I first wanna 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 μ defined only on the compact subsets of X. I wont 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 μ. Then μ can be shown to be finitely additive, subadditive and monotone on compact subsets. We then define the function $μ_*(U)=sup\{μ(K)|\text{ K is compact }, K⊂U\}$ defined on the open subsets of X, and then finally define an outer measure defined on all subsets, given by $μ^*(A)=inf\{μ_*(U)| \text{ U is open }, A⊂U\}$. 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 $μ^*=μ$ on all compact subsets of X. Hence let K be a compact subset of X. I have already shown that $μ^*(K) ≥ μ(K)$ using the following proof:
For all $\epsilon > 0$ there exists some open subset U such that K⊂U and such that $μ^*(K) + \epsilon ≥ μ_*(U)$. Additionally $μ_*(U) ≥ μ(K)$ hence $μ^*(K) + \epsilon ≥ μ(K)$. This holds for all $\epsilon > 0$ hence $μ^*(K) ≥ μ(K)$.
Hence I only need to show that $μ^*(K) ≤ μ(K)$. 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⊂U and such that $μ_*(U) ≤ μ(C) + \epsilon$. Since $μ^*(K) ≤ μ_*(U)$ it follows that $μ^*(K)≤μ(C)+\epsilon$. Now the problem is of course that I have to show that $μ^*(K) ≤ μ(K) + \epsilon$ and I don't know how $μ(C)$ and $μ(K)$ relates. I tried constructing some open subset U such that $μ(C) ≤ μ(K) + \epsilon$ whenever C⊂U and K⊂U since then $μ^*(K) ≤ μ(K) + \epsilon$, but I couldn't manage to construct said U.
I really feel like I miss something very obvious and its really nagging me. Any help would be appreciated!