Measure, Integration & Real Analysis · Problem 2D.24

Integration — Sheldon Axler

Mathematics — Measure Theory · inner-measure

Problem

For

A \subseteq \mathbf{R}

, the quantity

\sup\{|F| : F \text{ is a closed bounded subset of } \mathbf{R} \text{ and } F \subseteq 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}