Chapter 1

A family of sets \(\mathcal{R} \subset \mathcal{P}(X)\) is called a ring if it is closed under finite unions and differences (i.e., if \(E_{1}, \ldots, E_{n} \in \mathcal{R}\), then \(\bigcup_{1}^{n} E_{j} \in \mathcal{R}\), and if \(E, F \in \mathcal{R}\), then \(E \backslash F \in \mathcal{R}\) ). A ring that is closed under countable unions is called a \(\sigma\)-ring. a. Rings (resp. \(\sigma\)-rings) are closed under finite (resp. countable) intersections. b. If \(\mathcal{R}\) is a ring (resp. \(\sigma\)-ring), then \(\mathcal{R}\) is an algebra (resp. \(\sigma\)-algebra) iff \(X \in \mathcal{R}\). c. If \(\mathcal{R}\) is a \(\sigma\)-ring, then \(\left\\{E \subset X: E \in \mathcal{R}\right.\) or \(\left.E^{c} \in \mathcal{R}\right\\}\) is a \(\sigma\)-algebra. d. If \(\mathcal{R}\) is a \(\sigma\)-ring, then \(\\{E \subset X: E \cap F \in \mathcal{R}\) for all \(F \in \mathcal{R}\\}\) is a \(\sigma\)-algebra.

Let \(\mathcal{M}\) be an infinite \(\sigma\)-algebra. a. M contains an infinite sequence of disjoint sets. b. \(\operatorname{card}(\mathcal{M}) \geq c\).

If \(\mathcal{M}\) is the \(\sigma\)-algebra generated by \(\mathcal{\varepsilon}\), then \(\mathcal{M}\) is the union of the \(\sigma\)-algebras generated by \(\mathcal{F}\) as \(\mathcal{F}\) ranges over all countable subsets of \(\mathcal{E}\). (Hint: Show that the latter object is a \(\sigma\)-algebra.)

If \(\mu_{1}, \ldots, \mu_{n}\) are measures on \((X, \mathcal{M})\) and \(a_{1}, \ldots, a_{n} \in[0, \infty)\), then \(\sum_{1}^{n} a_{j} \mu_{j}\) is a measure on \((X, \mathcal{M})\).

If \((X, \mathcal{M}, \mu)\) is a measure space and \(\left\\{E_{j}\right\\}_{1}^{\infty} \subset \mathcal{M}\), then \(\mu\left(\lim \inf E_{j}\right) \leq\) \(\liminf \mu\left(E_{j}\right) .\) Also, \(\mu\left(\lim \sup E_{j}\right) \geq \lim \sup \mu\left(E_{j}\right)\) provided that \(\mu\left(\bigcup_{1}^{\infty} E_{j}\right)<\) \(\infty\).

Given a measure space \((X, \mathcal{M}, \mu)\) and \(E \in \mathcal{M}\), define \(\mu_{E}(\mathcal{A})=\mu(A \cap E)\) for \(A \in \mathcal{M}\). Then \(\mu_{E}\) is a measure.

Let \((X, \mathcal{M}, \mu)\) be a finite measure space. a. If \(E, F \in \mathcal{M}\) and \(\mu(E \Delta F)=0\), then \(\mu(E)=\mu(F)\). b. Say that \(E \sim F\) if \(\mu(E \Delta F)=0\); then \(\sim\) is an equivalence relation on \(\mathcal{M}\). c. For \(E, F \in \mathcal{M}\), define \(\rho(E, F)=\mu(E \Delta F)\). Then \(\rho(E, G) \leq \rho(E, F)+\) \(\rho(F, G)\), and hence \(\rho\) defines a metric on the space \(\mathcal{M} / \sim\) of equivalence classes.

Let \((X, \mathcal{N}, \mu)\) be a measure space. A set \(E \subset X\) is called locally measurable if \(E \cap A \in \mathcal{M}\) for all \(A \in \mathcal{M}\) such that \(\mu(A)<\infty\). Let \(\widetilde{\mathcal{M}}\) be the collection of all locally measurable sets. Clearly \(\mathcal{M} \subset \widetilde{\mathcal{M}}\); if \(\mathcal{M}=\widetilde{\mathcal{M}}\), then \(\mu\) is called saturated. a. If \(\mu\) is \(\sigma\)-finite, then \(\mu\) is saturated. b. \(\widetilde{\mathcal{M}}\) is a \(\sigma\)-algebra. c. Define \(\tilde{\mu}\) on \(\tilde{\mathcal{M}}\) by \(\tilde{\mu}(E)=\mu(E)\) if \(E \in \mathcal{M}\) and \(\tilde{\mu}(E)=\infty\) otherwise. Then \(\widetilde{\mu}\) is a saturated measure on \(\tilde{\mathcal{M}}\), called the saturation of \(\mu\). d. If \(\mu\) is complete, so is \(\tilde{\mu}\). e. Suppose that \(\mu\) is semifinite. For \(E \in \widetilde{\mathcal{M}}\), define \(\mu(E)=\sup \\{\mu(A): A \in\) \(\mathcal{M}\) and \(A \subset E\\}\). Then \(\mu\) is a saturated measure on \(\tilde{\mathcal{M}}\) that extends \(\mu\). f. Let \(X_{1}, X_{2}\) be disjoint uncountable sets, \(X=X_{1} \cup X_{2}\), and \(\mathcal{M}\) the \(\sigma\)-algebra of countable or co-countable sets in \(X\). Let \(\mu_{0}\) be counting measure on \(\mathcal{P}\left(X_{1}\right)\), and define \(\mu\) on \(\mathcal{M}\) by \(\mu(E)=\mu_{0}\left(E \cap X_{1}\right)\). Then \(\mu\) is a measure on \(\mathcal{M}\), \(\widetilde{\mathcal{M}}=\mathcal{P}(X)\), and in the notation of parts (c) and (e), \(\widetilde{\mu} \neq \underline{\mu}\).

If \(\mu^{*}\) is an outer measure on \(X\) and \(\left\\{A_{j}\right\\}_{1}^{\infty}\) is a sequence of disjoint \(\mu^{*}\) measurable sets, then \(\mu^{*}\left(E \cap\left(\bigcup_{1}^{\infty} A_{j}\right)\right)=\sum_{1}^{\infty} \mu^{*}\left(E \cap A_{j}\right)\) for any \(E \subset X .\)

Let \(\mathcal{A} \subset \mathcal{P}(X)\) be an algebra, \(\mathcal{A}_{\sigma}\) the collection of countable unions of sets in \(\mathcal{A}\), and \(\mathcal{A}_{\sigma \sigma}\) the collection of countable intersections of sets in \(\mathcal{A}_{\sigma}\). Let \(\mu_{0}\) be a premeasure on \(\mathcal{A}\) and \(\mu^{*}\) the induced outer measure. a. For any \(E \subset X\) and \(\epsilon>0\) there exists \(A \in \mathcal{A}_{\sigma}\) with \(E \subset A\) and \(\mu^{*}(A) \leq\) \(\mu^{*}(E)+\epsilon\). b. If \(\mu^{*}(E)<\infty\), then \(E\) is \(\mu^{*}\)-measurable iff there exists \(B \in A_{\sigma \delta}\) with \(E \subset B\) and \(\mu^{*}(B \backslash E)=0\). c. If \(\mu_{0}\) is \(\sigma\)-finite, the restriction \(\mu^{*}(E)<\infty\) in (b) is superfluous.