Question

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.

Short answer

Part a: Use countable unions. Part b: Measurability involves a zero-measure difference. Part c: \(\sigma\)-finiteness removes the finiteness restriction.

Step-by-step solution

Understanding the Problem

We need to show different properties related to the induced outer measure \(\mu^{*}\) from a premeasure \(\mu_{0}\) on an algebra \(\mathcal{A}\). Our task includes proving statements about sets \(E\) in terms of their inclusion in specific collections, \(\mathcal{A}_{\sigma}\) for part (a), measurability conditions in terms of \(\mathcal{A}_{\sigma\delta}\) for part (b), and considerational subtleties around \(\sigma\)-finiteness for part (c).

Proving Part (a)

The goal is to show that for any \(E \subset X\) and \(\epsilon>0\), there exists \(A \in \mathcal{A}_{\sigma}\) with \(E \subset A\) and \(\mu^{*}(A) \leq \mu^{*}(E)+\epsilon\). This is a standard result about outer measures where the set \(A\) can be constructed as a countable union of sets from \(\mathcal{A}\) whose measure approximates that of \(E\). This follows from the definition of \(\mu^{*}(E)\) as the infimum over all covers of \(E\) from \(\mathcal{A}\).

Proving Part (b)

We need to show that if \(\mu^{*}(E)<\infty\), then \(E\) is \(\mu^{*}\)-measurable if and only if there exists \(B \in \mathcal{A}_{\sigma\delta}\) such that \(E \subset B\) and \(\mu^{*}(B \backslash E)=0\). This uses the definition of \(\mu^{*}\)-measurability, where a set \(E\) is \(\mu^{*}\)-measurable if for any \(S\), \(\mu^{*}(S) = \mu^{*}(S \cap E) + \mu^{*}(S \cap E^c)\). The construction of \(B\) leverages \(\mathcal{A}_{\sigma\delta}\)'s closure properties.

Proving Part (c)

In this part, we demonstrate that if \(\mu_{0}\) is \(\sigma\)-finite, the condition \(\mu^{*}(E)<\infty\) in part (b) is not needed. \(\sigma\)-finiteness implies there exists a countable cover of \(X\) by sets with finite measure, simplifying measurability considerations and allowing \(E\) to be \(\mu^{*}\)-measurable without restriction to finiteness.

Key concepts

Algebra

In measure theory, an algebra is a collection of sets closed under the operations of union, intersection, and complement relative to a universal set. Consider it a well-behaved set system that ensures each operation results in another set in the same collection. A universal set here is a set containing all objects of interest in a particular discussion.

• Union: If sets A and B are in the algebra, then A ∪ B is also in it.
• Intersection: If sets A and B are in the algebra, A ∩ B also belongs to the algebra.
• Complement: If set A is in the algebra, then the complement of A, relative to the universal set, is in the algebra.

An algebra is a foundational structure that provides a framework on which premeasures and, subsequently, measures can be defined.

Premeasure

A premeasure is a function defined on an algebra that helps construct a measure. It provides a way to assign "size" or "volume" to sets in an algebra. While not yet a full measure, it offers the building blocks necessary to create one.
Let's break this down:

• A premeasure \( \mu_{0} \) is defined on the algebra \( \mathcal{A} \).
• \( \mu_{0} \) must be non-negative and include a measure of zero for the empty set: \( \mu_{0}(\emptyset) = 0 \).
• It is finitely additive, meaning if two disjoint sets \( A \) and \( B \) are in \( \mathcal{A} \), then \( \mu_{0}(A \cup B) = \mu_{0}(A) + \mu_{0}(B) \).

The goal of premeasures is to enable the construction of outer measures, thus preparing the landscape for defining full measures.

Sigma-finiteness

Sigma-finiteness is a property of premeasures or measures that relates to their manageability. A premeasure \( \mu_{0} \) is said to be \( \sigma \)-finite if the whole space can be covered by a countable union of sets with finite premeasure.
Why is this important?

• Ensures tractability: It makes analyzing infinite spaces feasible by breaking them into manageable parts.
• Extends measurability: When \( \mu_{0} \) is \( \sigma \)-finite, assumptions of finiteness in proofs can often be removed.

This property can simplify complex measure-theoretic arguments by allowing more flexibility in how sets and their measures are considered.

Measurability

Measurability in the context of outer measures, like \( \mu^{*} \), refers to the idea of a set being "well-behaved" with regard to that measure. A set is \( \mu^{*} \)-measurable if certain criteria are met that facilitate consistent calculations with it.
Here's how it works:

• Given a set \( E \), it is \( \mu^{*} \)-measurable if for any other set \( S \), the outer measure satisfies \( \mu^{*}(S) = \mu^{*}(S \cap E) + \mu^{*}(S \cap E^c) \).
• This condition ensures that measuring the part of \( S \) inside \( E \) and the part outside \( E \) gives the complete measure when combined.

Establishing measurability often involves finding a "nice" cover, such as a set \( B \) whose difference from \( E \) has negligible measure, making calculations straightforward and consistent.