Can Nonmeasurable Sets Have Finite Outer Measures?

  • Thread starter Thread starter cerenacerenb
  • Start date Start date
  • Tags Tags
    Proof Set
Click For Summary
SUMMARY

A set E with a finite outer measure can be nonmeasurable, as demonstrated in the discussion. If E is not measurable, there exists an open set O that contains E, which also has a finite outer measure. The key conclusion is that the difference between the outer measure of O and E, denoted as m*(O~E), is greater than the outer measure of O minus the outer measure of E, expressed as m*(O) - m*(E). This establishes a critical relationship between measurable and nonmeasurable sets in measure theory.

PREREQUISITES
  • Understanding of measure theory concepts, specifically outer and inner measures.
  • Familiarity with the definitions of measurable and nonmeasurable sets.
  • Knowledge of open sets in the context of topology.
  • Basic mathematical proof techniques, particularly in set theory.
NEXT STEPS
  • Study the properties of outer measures in detail, focusing on finite outer measures.
  • Explore the implications of the Lebesgue measure on measurable and nonmeasurable sets.
  • Investigate the role of open sets in measure theory and their relationship to measurable sets.
  • Learn about the construction of nonmeasurable sets, such as the Vitali set.
USEFUL FOR

Mathematicians, students of advanced calculus or real analysis, and anyone interested in the foundations of measure theory and its implications in mathematical analysis.

cerenacerenb
Messages
2
Reaction score
0
1. Let a set E have a finite outer measure. Show that if E is not measurable, then there is an open set O containing the set that has finite outer measure and for which m*(O~E) > m*(O)-m*(E).
 
Physics news on Phys.org
Any idea how to takle this proof?
 
You waited two whole minutes for a response before bumping?

A set is measurable if and only if both outer and inner measure are finite and equal. The inner measure is non-negative and less than the outer measure so since the outer measure is finite, so is the inner measure. That means that, since this set is not measurable, the inner measure must be less than the outer measure. What if O is an open set, containing E, that is measurable?
 

Similar threads

Nonmeasurable Set with Finite Outer Measure
  • · Replies 2 ·
Replies
2
Views
2K
Can Measurable Sets Be Written as Disjoint Union of Countable Collection?
  • · Replies 2 ·
Replies
2
Views
2K
Proving Finite Outer Measure Inequality
  • · Replies 1 ·
Replies
1
Views
2K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Prove that every subset ##B## of ##A=\{1,...,n\}## is finite
  • · Replies 4 ·
Replies
4
Views
1K
Egoroff's Theorem: Finite Measure Convergence
  • · Replies 3 ·
Replies
3
Views
3K
Is the Monotone Class Theorem Applicable to the Borel Sets?
  • · Replies 8 ·
Replies
8
Views
2K
Proving Compact Set Exists with m(E)=c
  • · Replies 11 ·
Replies
11
Views
3K
Problem on a set which is a subset of a finite set
  • · Replies 3 ·
Replies
3
Views
2K
If O is an event space, show for a finite number of events--
  • · Replies 5 ·
Replies
5
Views
2K