# Inner Measure not a Measure on Power Set, and Equals Outer Measure for Lebesgue Measurable set

Is there any advice on this problem:

For $$A \subset \mathbf{R},$$ the quantity $$\sup \{|E|: E$$ is a closed bounded subset of $$\mathbf{R}$$ and $$E \subset A\}$$ is called the inner measure of $$A .$$

(a) Show that if $$A$$ is a Lebesgue measurable subset of $$\mathbf{R}$$, then the inner measure of $$A$$ equals the outer measure of $$A .$$

(b) Show that inner measure is not a measure on the $$\sigma$$ -algebra of all subsets of $$\mathbf{R}$$

For (b) would the fact that $$\sup(\varnothing)=-\infty$$ be of use?

The reason why this holds is because there is no "perfect" measure. In Axler's text, this cannot be a measure since it is not a measure on all sets of $$\mathbb{R}$$
• The reason why this holds is because inner measure is not countably additive on the sigma-algebra of all subsets of $\mathbf{R}$. Nov 4, 2020 at 7:14