Finer Properties of Measurable Sets

Show that a set $E$ is measurable if for each $\epsilon > 0$, there is a closed set $F$ and open set $\mathcal{O}$ for which $F \subseteq E \subseteq \mathcal{O}$ and $m(\mathcal{O} \sim F) < \epsilon$.

Assume that $m^{*}(E) < \infty$. Show that there is a $G_{\delta }$ set $G$ that contains $E$ and $m(G) = m^{*}(E)$. Show that $E$ is measurable if and only if there is a $G_{\delta }$ set that contains $G$ and $m^{*}(G \sim E) = 0$.

Assume that $m^{*}(E) < \infty$. Show that if $E$ is not measurable, then there is an open set $\mathcal{O}$ containing $E$ that has finite outer-measure and for which

Let $E$ have finite outer-measure. Show that $E$ is measurable if and only if for each open, bounded interval $(a, b)$,

For any set $A$, define $m^{* *}(A) \in [0, \infty ]$ by

How is this set-function $m^{* *}$ related to outer-measure $m^{*}$ ?

For any set $A$, define $m^{* * *}(A) \in [0, \infty ]$ by

How is this set-function $m^{* * *}$ related to outer-measure $m^{*}$ ?