Problem 21
Let \(\mu^{*}\) be an outer measure induced from a premeasure and \(\bar{\mu}\) the restriction of \(\mu^{*}\) to the \(\mu^{*}\)-measurable sets. Then \(\bar{\mu}\) is saturated.
Problem 24
Let \(\mu\) be a finite measure on \((X, \mathcal{M})\), and let \(\mu^{*}\) be the outer measure induced by \(\mu\). Suppose that \(E \subset X\) satisfies \(\mu^{*}(E)=\mu^{*}(X)\) (but not that \(E \in \mathcal{M}\) ). a. If \(A, B \in \mathcal{M}\) and \(A \cap E=B \cap E\), then \(\mu(A)=\mu(B)\). b. Let \(\mathcal{M}_{E}=\\{A \cap E: A \in \mathcal{M}\\}\), and define the function \(\nu\) on \(\mathcal{M}_{E}\) defined by \(\nu(A \cap E)=\mu(A)\) (which makes sense by (a)). Then \(\mathcal{M}_{E}\) is a \(\sigma\)-algebra on \(E\) and \(\nu\) is a measure on \(\mathrm{M}_{E}\).
Problem 30
If \(E \in \mathcal{L}\) and \(m(E)>0\), for any \(\alpha<1\) there is an open interval \(I\) such that \(m(E \cap I)>\alpha m(I)\).
Problem 32
Suppose \(\left\\{\alpha_{j}\right\\}_{1}^{\infty} \subset(0,1)\). a. \(\prod_{1}^{\infty}\left(1-\alpha_{j}\right)>0\) iff \(\sum_{1}^{\infty} \alpha_{j}<\infty\). (Compare \(\sum_{1}^{\infty} \log \left(1-\alpha_{j}\right)\) to \(\left.\sum \alpha_{j} .\right)\) b. Given \(\beta \in(0,1)\), exhibit a sequence \(\left\\{\alpha_{j}\right\\}\) such that \(\prod_{1}^{\infty}\left(1-\alpha_{j}\right)=\beta\).
Problem 33
There exists a Borel set \(A \subset[0,1]\) such that \(0
