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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?

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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 $\mathbb{R}$

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  • $\begingroup$ The reason why this holds is because inner measure is not countably additive on the sigma-algebra of all subsets of $\mathbf{R}$. $\endgroup$
    – Sheldon Axler
    Nov 4, 2020 at 7:14

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