Let $(X,\mathcal{U})$ be a locally compact Hausdorff space, and $\mathcal B$ the Borel $\sigma$-Algebra respectively. We call a measure $\mu : \mathcal B \to [0, \infty]$ a Borel measure, if for every compact set $K\in \mathcal B$ it follows $\mu(B) < \infty$. Further $\mu$ is called inner regular if $\forall A \in \mathcal B\quad \mu(A) = \sup\{\mu(K)| K \text{ is compact & } K \subset A\}$. Now we defined a Radon measure as an inner regular Borel measure. Is there an example of a Borel measure which is not inner regular and hence not Radon?
Hopefully I am not creating duplicates here. Due to the fact that there are some different definitions I haven't found a proper answer.
Edit: I've read something about Dieudonné's Measure which is highly not trivial, so is there a simpler example?