Here are two definitions for the Lebesgue inner measure (below $\mu$ and $\mu^*$ are the Lebesgue measure and Lebesgue outer measure respectively):
Definition 1: the Lebesgue inner measure $\mu_*:\mathcal{P}(\mathbb{R}^d)\to\mathbb{R}$ is defined as $$\mu_*:A\mapsto \sup\left\{\mu(C) : \text{ $C$ is a closed, measurable subset of $A$} \right\}.$$
Definition 2: the Lebesgue inner measure $\mu_*:\left\{\text{bounded subsets of $\mathbb{R}^d$}\right\}\to\mathbb{R}$ is defined as $$\mu_*:A\mapsto \mu(E)-\mu^*(E\setminus A)$$ for any elementary set $E$ containing $A$.
Are the two definitions above equivalent (for bounded sets)?
Is any of the above functions superadditive?
