Measure associated to locally integrable function is regular

Suppose $f \in L^{1}_\text{loc}(\mathbb{R}^{d})$ with $f \geq 0$ and create the measure $\mu$ such that $d\mu = f\, dx$ where $dx$ is the standard Lebesgue measure. Must $\mu$ be a regular measure?

Since $f$ is locally integrable, $\mu(K) < \infty$ for any compact $K$, but what about inner and outer regularity?