The Carath'{e}odory-Hahn Theorem

Let $S \subseteq 2^{X}$ and $\mu^{*}: 2^{X} \rightarrow [0, \infty ]$ be the outer-measure induced by $\mu : S \rightarrow [0, \infty ]$. Show that for a set $A \subseteq X$ to be measurable with respect to $\mu^{*}$ it suffices to verify that

not for all of the subsets $E$ of $X$. This was the definition of measurable set used by Lebesgue for the set-function length defined on the collection of open, bounded intervals in $\mathbb {R}$.

Show that a non-negative set-function on a $\sigma$-algebra is a measure if and only if it is a premeasure.

Let $\mathbb {Q}$ be the set of rational numbers and $\mathcal{S}$ the collection of all finite unions of intervals of the form $(a, b] \cap \mathbb {Q}$, where $a, b \in \mathbb {Q}$ and $a \leq b$. Define $\mu (a, b] = \infty$ if $a < b$ and $\mu (\emptyset ) = 0$. Show that $\mathcal{S}$ is closed with respect to relative complements and $\mu : \mathcal{S} \rightarrow [0, \infty ]$ is a premeasure. Then show that the extension of $\mu$ to the smallest $\sigma$-algebra containing $\mathcal{S}$ is not unique.

Verify the assertions made in this section's first and second example.

Show that the three collections of sets in this section's third example are semi-rings.

Let $\mathcal{S}$ be a $\sigma$-algebra of subsets of $X$ and $\mu : \mathcal{S} \rightarrow [0, \infty ]$ be a measure. Let $\bar{\mu }: \overline{\mathcal{M}} \rightarrow [0, \infty ]$ be the Carathéodory measure induced by $\mu$. Show that $\mathcal{S}$ is a subcollection of $\overline{\mathcal{M}}$ and it may be a proper subcollection.

Show that Lebesgue on $\mathbb {R}$ is the Carathéodory extension of the premeasure Lebesgue measure on the semi-ring of Borel sets.