Section 2.3

The $\sigma$ -Algebra of Lebesgue Measurable Sets

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Problem 2.12

Prove that if a $\sigma$ -algebra of subsets of $\mathbb {R}$ contains intervals of the form ( $a, \infty$ ), then it contains all intervals.

Problem 2.13

(i)(ii)(iii)

Show that

(i)

the translate of an $F_{\sigma }$ set is also $F_{\sigma }$ ,

(ii)

the translate of a $G_{\delta }$ set is also $G_{\delta }$ ,

(iii)

the translate of a set of measure zero also has measure zero.

Problem 2.14

Show that if a set $E$ has positive outer-measure, then there is a bounded subset of $E$ that also has positive outer-measure.

Problem 2.15

Show that if $m(E) < \infty$ and $\epsilon > 0$ , then $E$ is the disjoint union of a finite number of measurable sets, each of which has measure at most $\epsilon$ .