Sum of Lebesgue inner and outer measures is a non-regular metric outer measure on $\mathbb{R}$

Question

An outer measure $\mu$ on a metric space $(X,d)$ is metric if $\forall A, B \subset X: d(A,B) >0 \implies \mu(A\cup B) = \mu(A) + \mu(B)$. I came across the following example that not every metric outer measure is regular: $\mu = \frac{1}{2}(m_{*} + m^{*})$, where $m_{*}, m^{*}$ are the Lebesgue inner and outer measures respectively. I'm stuck at proving that $\mu$ is $\sigma$-subadditive. Any help is appreciated.