Measures Induced by an Outer-Measure

Show that in the construction of Lebesgue measure on $\mathbb {R}$, the outer-measure is unchanged if covers by open, bounded intervals are replaced by covers by general, bounded intervals.

Let $\mu^{*}: 2^{X} \rightarrow [0, \infty ]$ be an outer-measure. Let $A \subseteq X,\left\{ E_{k}\right\}_{k = 1}^{\infty }$ be a countable, disjoint collection of measurable sets and $E = \bigcup_{k = 1}^{\infty } E_{k}$. Show that

Show that any measure that is induced by an outer-measure is complete.

Define $\mu : 2^{\mathbb {R}} \rightarrow \mathbb {R}$ by defining $\mu (A)$ to be the number of integers in $A$. Determine the outer-measure $\mu^{*}$ induced by $\mu$ and the $\sigma$-algebra of measurable sets.