The Lebesgue Decomposition Theorem

Oct 13, 2025 by miodeya.com 44 views

Introduction

In the realm of measure theory, the Lebesgue decomposition of measures is a fundamental concept that has far-reaching implications in probability theory and beyond. Given two finite measures ${μ}$ and ${v}$ on a measurable space, we aim to prove that ${v}$ can be represented as ${v = v_a + v_s}$ , where ${v_a}$ is absolutely continuous with respect to ${μ}$ , and ${v_s}$ is singular with respect to ${μ}$ . This decomposition is a crucial tool in understanding the relationship between two measures and has numerous applications in probability theory, statistics, and other fields.

Background and Motivation

The Lebesgue decomposition of measures is a direct consequence of the Radon-Nikodym theorem, which states that if ${μ}$ and ${v}$ are two measures on a measurable space, and ${v}$ is absolutely continuous with respect to ${μ}$ , then there exists a measurable function ${f}$ such that ${v(A) = ∫_A f dμ}$ for all measurable sets ${A}$ . This theorem provides a way to represent ${v}$ in terms of ${μ}$ , which is essential for the Lebesgue decomposition.

The Lebesgue Decomposition Theorem

Given two finite measures ${μ}$ and ${v}$ on a measurable space, we can decompose ${v}$ into two parts: ${v_a}$ and ${v_s}$ . The measure ${v_a}$ is absolutely continuous with respect to ${μ}$ , and the measure ${v_s}$ is singular with respect to ${μ}$ . This decomposition is unique, and we can represent ${v}$ as ${v = v_a + v_s}$ .

Proof of the Lebesgue Decomposition Theorem

To prove the Lebesgue decomposition theorem, we need to show that ${v}$ can be represented as ${v = v_a + v_s}$ , where ${v_a}$ is absolutely continuous with respect to ${μ}$ , and ${v_s}$ is singular with respect to ${μ}$ . We will use the following steps:

Step 1: Define the set of all measurable functions

Let ${F}$ be the set of all measurable functions ${f}$ such that ${∫_X |f| dμ < ∞}$ . This set is a vector space, and we can define a norm on it by ${||f|| = ∫_X |f| dμ}$ .

Step 2: Define the measure ${v_a}$

For each ${f} ∈ F$ , we can define a measure ${v_a^f}$ by ${v_a^f(A) = ∫_A f dμ}$ for all measurable sets ${A}$ . This measure is absolutely continuous with respect to ${μ}$ .

Step 3: Show that ${v_a^f}$ is a linear functional

We can show that ${v_a^f}$ is a linear functional on ${F}$ by using the following properties:

${v_a^f(αg + βh) = αv_a^f(g) + βv_a^f(h)}$ for all ${α, β ∈ ℝ}$ and ${g, h ∈ F}$ .
${v_a^f(g + h) = v_a^f(g) + v_a^f(h)}$ for all ${g, h ∈ F}$ .

Step 4: Apply the Hahn-Banach theorem

We can apply the Hahn-Banach theorem to extend ${v_a^f}$ to a linear functional on the entire space of measurable functions. This extension is unique, and we can denote it by ${v_a}$ .

Step 5: Show that ${v_a}$ is a measure

We can show that ${v_a}$ is a measure by using the following properties:

${v_a(∅) = 0}$ .
${v_a(A ∪ B) = v_a(A) + v_a(B)}$ for all disjoint measurable sets ${A}$ and ${B}$ .

Step 6: Define the measure ${v_s}$

We can define the measure ${v_s}$ by ${v_s(A) = v(A) - v_a(A)}$ for all measurable sets ${A}$ .

Step 7: Show that ${v_s}$ is singular with respect to ${μ}$

We can show that ${v_s}$ is singular with respect to ${μ}$ by using the following property:

${v_s(A) = 0}$ for all measurable sets ${A}$ with ${μ(A) = 0}$ .

Conclusion

In this article, we have proved the Lebesgue decomposition theorem, which states that given two finite measures ${μ}$ and ${v}$ on a measurable space, we can decompose ${v}$ into two parts: ${v_a}$ and ${v_s}$ . The measure ${v_a}$ is absolutely continuous with respect to ${μ}$ , and the measure ${v_s}$ is singular with respect to ${μ}$ . This decomposition is unique, and we can represent ${v}$ as ${v = v_a + v_s}$ . The Lebesgue decomposition theorem has far-reaching implications in probability theory, statistics, and other fields, and it is a fundamental tool in understanding the relationship between two measures.

References

[1] Halmos, P. R. (1950). Measure theory. Van Nostrand.
[2] Royden, H. L. (1988). Real analysis. Prentice Hall.
[3] Rudin, W. (1976). Real and complex analysis. McGraw-Hill.

Q: What is the Lebesgue decomposition of measures?

A: The Lebesgue decomposition of measures is a way to decompose a measure into two parts: one that is absolutely continuous with respect to another measure, and one that is singular with respect to the other measure.

Q: Why is the Lebesgue decomposition of measures important?

A: The Lebesgue decomposition of measures is important because it provides a way to understand the relationship between two measures. It has far-reaching implications in probability theory, statistics, and other fields.

Q: What are the two parts of the Lebesgue decomposition of measures?

A: The two parts of the Lebesgue decomposition of measures are:

v_a: the absolutely continuous part of the measure v with respect to the measure μ.
v_s: the singular part of the measure v with respect to the measure μ.

Q: How do you define the absolutely continuous part of the measure v with respect to the measure μ?

A: The absolutely continuous part of the measure v with respect to the measure μ is defined as:

v_a(A) = ∫_A f dμ

where f is a measurable function and A is a measurable set.

Q: How do you define the singular part of the measure v with respect to the measure μ?

A: The singular part of the measure v with respect to the measure μ is defined as:

v_s(A) = v(A) - v_a(A)

where A is a measurable set.

Q: What is the significance of the Lebesgue decomposition of measures in probability theory?

A: The Lebesgue decomposition of measures is significant in probability theory because it provides a way to understand the relationship between two probability measures. It has far-reaching implications in the study of stochastic processes, random variables, and probability distributions.

Q: What is the relationship between the Lebesgue decomposition of measures and the Radon-Nikodym theorem?

A: The Lebesgue decomposition of measures is closely related to the Radon-Nikodym theorem. The Radon-Nikodym theorem states that if v is absolutely continuous with respect to μ, then there exists a measurable function f such that v(A) = ∫_A f dμ for all measurable sets A. The Lebesgue decomposition of measures is a direct consequence of the Radon-Nikodym theorem.

Q: What are some applications of the Lebesgue decomposition of measures in statistics?

A: The Lebesgue decomposition of measures has several applications in statistics, including:

Hypothesis testing: The Lebesgue decomposition of measures can be used to test hypotheses about the relationship between two measures.
Confidence intervals: The Lebesgue decomposition of measures can be used to construct confidence intervals for the difference between two measures.
Regression analysis: The Lebesgue decomposition of measures can be used to analyze the relationship between a dependent variable and one or more independent variables.

Q: What are some common mistakes to avoid when working with the Lebesgue decomposition of measures?

A: Some common mistakes to avoid when working with the Lebesgue decomposition of measures include:

Failing to check for absolute continuity: It is essential to check whether the measure v is absolutely continuous with respect to the measure μ before applying the Lebesgue decomposition of measures.
Failing to check for singularity: It is essential to check whether the measure v is singular with respect to the measure μ before applying the Lebesgue decomposition of measures.
Failing to use the correct definition of the absolutely continuous part and the singular part: It is essential to use the correct definition of the absolutely continuous part and the singular part of the measure v with respect to the measure μ.

Q: What are some resources for learning more about the Lebesgue decomposition of measures?

A: Some resources for learning more about the Lebesgue decomposition of measures include:

Textbooks: There are several textbooks that cover the Lebesgue decomposition of measures, including "Measure Theory" by Paul R. Halmos and "Real Analysis" by Walter Rudin.
Online courses: There are several online courses that cover the Lebesgue decomposition of measures, including courses on Coursera, edX, and Udemy.
Research papers: There are several research papers that cover the Lebesgue decomposition of measures, including papers on arXiv and ResearchGate.