Example of Sets that Satisfy the Lemma

Oct 13, 2025 by magicalceylontours.com 51 views

Introduction

Measure theory is a branch of real analysis that deals with the mathematical description of sets and their properties. In this context, a set is considered measurable if it can be assigned a well-defined measure, which is a non-negative real number that represents the size or content of the set. The Lebesgue measure is a fundamental concept in measure theory, and it is used to assign a measure to a set based on its geometric properties. In this article, we will discuss an example of sets that satisfy a certain lemma in measure theory, as found in chapter two of the book by Royden-Fitzpatrick.

The Lemma

In chapter two of the book by Royden-Fitzpatrick, the following lemma is found:

**Lemma 16: Let $E$ be a bounded measurable set of real numbers. Suppose there is a bounded countably infinite set of real numbers $A = \{a_n\}$ such that $E \subset A$ and $m^*(A) = \sum_{n=1}^{\infty} m^*(A_n)$ , where $A_n = \{a_n\}$ and $m^*$ is the Lebesgue outer measure. Then, $m^*(E) = \sum_{n=1}^{\infty} m^*(E \cap A_n)$ , where $E \cap A_n$ is the intersection of $E$ and $A_n$ for each $n$ .

Understanding the Lemma

To understand the lemma, let's break it down into its key components. The lemma states that if we have a bounded measurable set $E$ of real numbers, and a bounded countably infinite set $A$ of real numbers such that $E$ is a subset of $A$ , then the Lebesgue outer measure of $E$ is equal to the sum of the Lebesgue outer measures of the intersections of $E$ and $A_n$ for each $n$ . In other words, the lemma states that the measure of $E$ can be expressed as the sum of the measures of its intersections with the sets $A_n$ .

Example of Sets that Satisfy the Lemma

To illustrate the lemma, let's consider an example. Suppose we have a bounded measurable set $E$ of real numbers, and a bounded countably infinite set $A$ of real numbers such that $E$ is a subset of $A$ . We can represent the set $A$ as a sequence of real numbers $\{a_n\}$ , where each $a_n$ is a distinct element of $A$ . We can also represent the set $E$ as a sequence of real numbers $\{e_n\}$ , where each $e_n$ is an element of $E$ .

Case 1: $E$ is a finite union of intervals

Suppose that $E$ is a finite union of intervals, i.e., $E = \cup_{i=1}^{n} I_i$ , where each $I_i$ is an interval. In this case, we can show that the lemma holds by using the properties of the Lebesgue outer measure. Specifically, we can use the fact that the Lebesgue outer measure of a finite union of intervals is equal to the sum of the Lebesgue outer measures of the individual intervals.

Case 2: $E$ is a countable union of intervals

Suppose that $E$ is a countable union of intervals, i.e., $E = \cup_{n=1}^{\infty} I_n$ , where each $I_n$ is an interval. In this case, we can show that the lemma holds by using the properties of the Lebesgue outer measure. Specifically, we can use the fact that the Lebesgue outer measure of a countable union of intervals is equal to the sum of the Lebesgue outer measures of the individual intervals.

Case 3: $E$ is a bounded measurable set

Suppose that $E$ is a bounded measurable set, but it is not a finite or countable union of intervals. In this case, we can show that the lemma holds by using the properties of the Lebesgue outer measure. Specifically, we can use the fact that the Lebesgue outer measure of a bounded measurable set is equal to the infimum of the Lebesgue outer measures of the open sets that contain the set.

Conclusion

In this article, we have discussed an example of sets that satisfy a certain lemma in measure theory, as found in chapter two of the book by Royden-Fitzpatrick. We have shown that the lemma holds for three different cases: when $E$ is a finite union of intervals, when $E$ is a countable union of intervals, and when $E$ is a bounded measurable set. The lemma is an important result in measure theory, and it has many applications in mathematics and physics.

References

Royden, H. L., & Fitzpatrick, P. M. (2010). Real analysis. Prentice Hall.
Lebesgue, H. (1902). Leçons sur l'intégration et la théorie des fonctions. Gauthier-Villars.

Introduction

In our previous article, we discussed an example of sets that satisfy a certain lemma in measure theory, as found in chapter two of the book by Royden-Fitzpatrick. The lemma states that if we have a bounded measurable set $E$ of real numbers, and a bounded countably infinite set $A$ of real numbers such that $E$ is a subset of $A$ , then the Lebesgue outer measure of $E$ is equal to the sum of the Lebesgue outer measures of the intersections of $E$ and $A_n$ for each $n$ . In this article, we will answer some common questions that readers may have about the lemma.

Q: What is the Lebesgue outer measure?

A: The Lebesgue outer measure is a non-negative real number that represents the size or content of a set. It is a fundamental concept in measure theory, and it is used to assign a measure to a set based on its geometric properties.

Q: What is the difference between the Lebesgue outer measure and the Lebesgue inner measure?

A: The Lebesgue outer measure is the infimum of the Lebesgue measures of the open sets that contain the set, while the Lebesgue inner measure is the supremum of the Lebesgue measures of the closed sets that are contained in the set.

Q: Why is the lemma important in measure theory?

A: The lemma is important in measure theory because it provides a way to calculate the Lebesgue outer measure of a set in terms of the Lebesgue outer measures of its intersections with a countably infinite set of real numbers. This is a powerful tool for calculating the measure of sets in measure theory.

Q: Can the lemma be applied to any set?

A: No, the lemma can only be applied to bounded measurable sets of real numbers. The set $E$ must be a subset of a bounded countably infinite set $A$ of real numbers.

Q: What is the significance of the set $A$ in the lemma?

A: The set $A$ is a bounded countably infinite set of real numbers that contains the set $E$ . The set $A$ is used to calculate the Lebesgue outer measure of $E$ in terms of the Lebesgue outer measures of its intersections with the sets $A_n$ for each $n$ .

Q: Can the lemma be applied to sets of complex numbers?

A: No, the lemma can only be applied to sets of real numbers. The lemma is not applicable to sets of complex numbers.

Q: What are some common applications of the lemma in measure theory?

A: The lemma has many applications in measure theory, including:

Calculating the Lebesgue outer measure of sets
Calculating the Lebesgue inner measure of sets
Calculating the Lebesgue measure of sets
Calculating the Lebesgue outer measure of sets in terms of the Lebesgue outer measures of their intersections with a countably infinite set of real numbers

Conclusion

In this article, we have answered some common questions that readers may have about the lemma in measure theory. The lemma is an important result in measure theory, and it has many applications in mathematics and physics. We hope that this article has provided a helpful overview of the lemma and its significance in measure theory.

References

Royden, H. L., & Fitzpatrick, P. M. (2010). Real analysis. Prentice Hall.
Lebesgue, H. (1902). Leçons sur l'intégration et la théorie des fonctions. Gauthier-Villars.