Problem 14
Give an example of a set which is Lebesgue measurable, but not Jordan measurable.
Problem 15
We say that a set \(\mathrm{A}\) is a set of o-uniqueness for a o-additive
measure \(m\) if
1) There is a o-additive extension of \(m\) defined on \(A\);
2) If \(\mu_{1}\) and \(\mu_{2}\) are two such extensions, then
\(\mu_{1}(A)=\mu_{2}(A)\).
Prove that the system of sets of o-uniqueness of a o-additive measure \(m\)
defined on a semiring \(\mathscr{S}_{m}\) coincides with the system of sets
which are Lebesgue measurable (with respect to \(\mathrm{m}\) ).
Hint. To show that every Lebesgue-measurable set \(\mathrm{A}\) is a set of
\(\sigma\) uniqueness for \(m\), choose any \(\mathrm{E}>0 .\) Then there is a set
\(\mathrm{B} \in \mathscr{R}=\mathscr{R}\left(\mathscr{S}_{\mathrm{m}}\right)\)
such that \(\mu^{*}(A \triangle B)
