Proving Equality of Characteristic Functions using Lebesgue Measure?

  • Thread starter Thread starter Pyroadept
  • Start date Start date
  • Tags Tags
    Integral
Click For Summary
SUMMARY

The discussion centers on proving that the characteristic functions χ_E and χ_F of two Lebesgue measurable sets E and F are equal almost everywhere if and only if the Lebesgue measure of their symmetric difference, µ(EΔF), equals zero. The characteristic function χ_E indicates membership in set E, while µ(EΔF) quantifies the measure of the set where E and F differ. The key takeaway is that equality of characteristic functions almost everywhere directly correlates with the measure of their symmetric difference being zero.

PREREQUISITES
  • Understanding of Lebesgue measure and measurable sets
  • Familiarity with characteristic functions in measure theory
  • Knowledge of symmetric differences in set theory
  • Basic concepts of real analysis and properties of functions
NEXT STEPS
  • Study the properties of Lebesgue measurable sets in detail
  • Explore the concept of characteristic functions and their applications
  • Learn about symmetric differences and their implications in measure theory
  • Investigate proofs involving almost everywhere convergence in real analysis
USEFUL FOR

Mathematics students, particularly those studying real analysis and measure theory, as well as educators looking to deepen their understanding of characteristic functions and Lebesgue measure.

Pyroadept
Messages
82
Reaction score
0

Homework Statement


For sets E,F \in L, show that χ_E = χ_F almost everywhere if and only if µ(EΔF) = 0


where χ_E is the characteristic function w.r.t. E
and µ(EΔF) is the lebesgue measure of the symmetric difference of E and F
and L is the set of lebesgue measurable sets


Homework Equations



A property about real numbers holds almost everywhere if the set of x where it fails to be true has Lebesgue measure 0.


The Attempt at a Solution



I'm really stuck on this. I'm not asking anyone to do it for me, but if anyone could please give me a point in the right direction, that would be great thanks!
 
Physics news on Phys.org
Isn't the set where the characteristic function of E and F differ equal to the symmetric difference of E and F?
 

Similar threads

Inverse image of Lebesgue set and Borel set
  • · Replies 8 ·
Replies
8
Views
3K
Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?
  • · Replies 10 ·
Replies
10
Views
2K
Proving Measurability and Integrability of a Function on a Product Space
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
3K
Properties of Lebesgue Integral
  • · Replies 5 ·
Replies
5
Views
1K
Prove that an Lebesgue integral exists iff the other exists
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad A concise "proof" of the Riemann-Lebesgue lemma
  • · Replies 1 ·
Replies
1
Views
2K
Lebesgue Integral: Practice Problem
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
3K
Proving the Existence of an Interval in a Lebesgue Measure Space
  • · Replies 1 ·
Replies
1
Views
1K