Understanding Lebesgue Measurable Functions and Sets: A Comprehensive Guide

  • Thread starter pivoxa15
  • Start date
  • Tags
    Measurable
In summary, a function is Lebesgue measurable if the preimage of any interval is a measurable set, while a set is Lebesgue measurable if it satisfies the condition L*(A)=L*(AE)+L*(AE^c). This condition ensures that the Lebesgue measure is additive, and it is a practical characterization introduced by Lebesgue. It is worth noting that almost all sets are Lebesgue measurable, and the use of the axiom of choice is necessary to construct a set that is not measurable. This concept is commonly found in sources on measure theory.
  • #1
pivoxa15
2,255
1

Homework Statement


What does it mean for a function to be lebesgue measurable?

What does it mean for a set to be lebesgue measureable?
 
Physics news on Phys.org
  • #2
First we define the notion of an outer measure L*, which is a very intuitive generalization of the length of an interval. The outer measure of a set S in R is the inf over all collections of intervals covering S of the total length of those intervals.

Then we define the Lebesgue measure L by restricting the domain of L* to sets that we call measurable:

A set E is said Lebesgue measurable if for any set A, we have

L*(A)=L*(AE)+L*(AE^c)

This strange looking condition is a more practical characterizations due to Carathéodory of the notion of measurability introduced by Lebesgue. In either case, the condition is there to insure that the measure L will be additive. I.e. for A, B disjoint, L(AuB)=L(A)+L(B). Actually, almost all sets are measurable, and to construct one that isn't, you must make explicit use of the axiom of choice. If we work with a set theory without the axiom of choice, all sets are Lebesgue-measurable.

And a function is measurable if the preimage of any interval is a meaurable set.
 
  • #3
Surely this is defined in *any* source on measure theory. It shouldn't be the job of this forum to read the bloody book on someone's behalf.
 

1. What is a Lebesgue measurable function?

A Lebesgue measurable function is a type of function used in measure theory, a branch of mathematics that deals with the concept of size or volume. A Lebesgue measurable function is a function that preserves the measure of sets, meaning that the measure of the set of points where the function takes a particular value is the same as the measure of the original set.

2. Why is understanding Lebesgue measurable functions important?

Understanding Lebesgue measurable functions is important because it allows us to analyze and measure more complicated sets and functions than traditional methods of integration. It is also a fundamental concept in many areas of mathematics, including analysis, probability, and statistics.

3. What is a Lebesgue measurable set?

A Lebesgue measurable set is a set that can be accurately measured using the Lebesgue measure. This measure is more general and flexible than traditional measures, allowing for more precise calculations of the size or volume of sets that may be difficult to measure using other methods.

4. How are Lebesgue measurable functions and sets related?

Lebesgue measurable functions and sets are closely related because measurable functions are defined using measurable sets. In particular, a function is Lebesgue measurable if and only if the pre-image of any measurable set under the function is also measurable.

5. What are some applications of Lebesgue measurable functions and sets?

Lebesgue measurable functions and sets have many applications in mathematics and other fields. They are used to solve problems in analysis, probability, statistics, and other branches of mathematics. They are also used in real-world applications such as signal processing, image analysis, and data compression.

Similar threads

Inverse image of Lebesgue set and Borel set
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
I Basic Probability Theory Question about Lebesgue measure
    • Set Theory, Logic, Probability, Statistics
Replies
14
Views
1K
I ##L^2## square integrable function Hilbert space
    • Calculus
Replies
11
Views
169
I Collection of finite unions of half-open intervals form an algebra
    • Topology and Analysis
Replies
3
Views
203
I Quantum particle's state in momentum eigenfunctions basis
    • Quantum Physics
Replies
14
Views
202
Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
A Functional Analysis for Physics in 2024
    • Topology and Analysis
Replies
11
Views
256
Proving Measurability and Integrability of a Function on a Product Space
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Equivalence of Lebesgue Measurability
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
A Karhunen–Loève theorem expansion random variables
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
104

Hot Threads

Recent Insights

Back
Top