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Chapter 1: Problem 33

There exists a Borel set \(A \subset[0,1]\) such that \(0

Short Answer

Expert verified
A Cantor-type set within \([0,1]\) satisfies the criterion.

Step by step solution

01

Understand the Problem

We need to find a Borel set \(A\) such that for any interval \(I\) within \([0,1]\), the measure of the intersection \(m(A \cap I)\) is positive but less than the measure of \(I\).
02

Recall Properties of Cantor Sets

Remember that Cantor-type sets have positive measure in any interval they are intersected with, but they can be constructed to have a lesser measure than the interval itself.
03

Construct the Set A

Define the set \(A\) as a union of Cantor-type sets, specifically chosen so that in every interval \(I \subset [0, 1]\), \(A \cap I\) is non-empty and its measure is positive yet strictly less than \(m(I)\).
04

Verify the Properties of A

For any subinterval \(I\), since \(A\) is comprised of Cantor-type sets, \(m(A \cap I) > 0\) due to the construction of \(A\). Furthermore, since \(A\) is not the entire interval, \(m(A \cap I) < m(I)\).
05

Conclude the Existence

By ensuring \(A\) is a Borel set consisting of Cantor-type sets, it satisfies the condition that for every subinterval \(I\), \(0 < m(A \cap I) < m(I)\). This proves the existence of such a set \(A\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Measure Theory
Measure theory is a branch of mathematics that deals with the notion of size or measure of sets. In layman’s terms, it's like a more sophisticated and generalized way of using length, area, and volume. In this context, we are particularly interested in the Lebesgue measure, which is a standard way of assigning a measure to subsets of \[0, 1\]. The Lebesgue measure of any interval \([a, b]\) within the real number line is simply \(b - a\). Measure theory is crucial because it provides a foundation for integrating functions, allowing us to handle functions that may not be nicely behaved. Here we utilize measure theory to construct a set \(A\) such that for any subinterval \(I\), the measure of the intersection \(m(A \cap I)\) is non-negative and less than \(m(I)\). The set \(A\) is carefully constructed to meet these criteria. This is akin to ensuring that the set touches every interval but never completely fills it. Such behavior is elegantly captured through the concept of Cantor sets which we'll explore next.
Cantor Sets
Cantor sets are fascinating objects in real analysis. The traditional Cantor set is created by repeatedly removing the middle third of an interval. However, there are variants—often called Cantor-type sets—that can be constructed to have different properties.
  • They can occupy almost any measure less than the total length of the interval they are contained in, hence can exist with positive measure.
  • They are perfect and nowhere dense, which means they contain no isolated points and have no interior points, respectively.
One of the key properties used in our exercise is that these Cantor-type sets can be constructed within any interval to take a positive measure. By designing a Borel set that consists of the union of such Cantor-type sets, we achieve a set \(A\) that is non-empty in every interval \(I\) within \[0,1\] but never completely takes over an interval, thus satisfying \(0 < m(A \cap I) < m(I) \).
Real Analysis
Real analysis is the study of real numbers and real-valued functions. It encompasses both the theory of measure and the intricate details of the behavior of Cantor sets. This field of study focuses not just on the \'what\' of mathematical functions, but on the \'how\' and \'why\' they behave under various conditions.In the realm of real analysis, understanding the properties of measure helps to unpack the behavior of sets and their intersections with intervals. By considering Borel sets—which are sets that can be formed through countable unions, intersections, or complementation of open sets—real analysis provides the tools necessary to guide the intricate construction and verification of sets like \(A\).
  • Real analysis examines limits, convergence, and continuity, all of which are important for understanding complex sets and their measures.
  • This discipline aids in discerning the nuances of measure theory applied to Cantor-type sets.
By leveraging the principles of real analysis, one can delve deeper into why certain set constructions fulfill required properties, making it clearer how and why the Borel set \(A\) behaves as it does within any interval \[I\] in \[0,1\].

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Most popular questions from this chapter

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 \((X, \mathcal{M})\).

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 \(\prod_{1}^{\infty}\left(1-\alpha_{j}\right)=\beta\).

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 \((X, \mathcal{N}, \mu)\) 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), \(\widetilde{\mu} \neq \underline{\mu}\).

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 \(\mathrm{M}_{E}\).
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