A Property of set with finite measure

Click For Summary
SUMMARY

The discussion centers on the property of a measurable set E with finite measure, specifically that it can be expressed as a disjoint union of a finite number of measurable subsets, each with a measure not exceeding a specified epsilon. The participant outlines their approach using open sets O_i that cover E and discusses the need to ensure the subsets are disjoint and their union equals E. A suggestion is made to utilize intersections and differences of the sets to achieve the desired properties.

PREREQUISITES
  • Understanding of measure theory concepts, particularly finite measure.
  • Familiarity with open sets and their properties in topology.
  • Knowledge of compactness and its implications in analysis.
  • Experience with set operations, including intersections and unions.
NEXT STEPS
  • Study the properties of finite measures in measure theory.
  • Learn about the role of compactness in topology and its applications.
  • Explore techniques for constructing disjoint unions of sets in measure theory.
  • Investigate the use of intersections and differences in set theory to manipulate measurable sets.
USEFUL FOR

Mathematics students, particularly those studying measure theory and topology, as well as educators seeking to understand the properties of measurable sets with finite measure.

ntsivanidis
Messages
4
Reaction score
0

Homework Statement



If E has finite measure and \epsilon>0, then E is the disjoint union of a finite number of measurable sets, each of which has measure at most \epsilon.

Homework Equations



The Attempt at a Solution


I proceeded by showing that by definition of measure, there is a finite group of open sets O_i that contain E, whose union has the same measure (and contains E). By taking their closure, by compactness each has an open cover of \epsilon neighborhoods of a finite number of points. The union of these, within each O_i and then across all O_i, contains E.

My problem is i)to ensure the finite number of subsets are disjoint, and ii) to ensure that the union of these sets is equal to E.

Thanks!
 

Attachments

  • Screen shot 2010-11-14 at 3.15.25 PM.png
    Screen shot 2010-11-14 at 3.15.25 PM.png
    24.4 KB · Views: 550
Last edited:
Physics news on Phys.org
Then why don't you take the intersections of your sets with E and then differences and intersections of the sets themselves?
 

Similar threads

My proof of the Geometry-Real Analysis theorem
  • · Replies 1 ·
Replies
1
Views
2K
Can Measurable Sets Be Written as Disjoint Union of Countable Collection?
  • · Replies 2 ·
Replies
2
Views
2K
Egoroff's Theorem: Finite Measure Convergence
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
Nonmeasurable Set with Finite Outer Measure
  • · Replies 2 ·
Replies
2
Views
2K
Proof for convergent sequences, limits, and closed sets?
  • · Replies 11 ·
Replies
11
Views
3K
Measure defined on Borel sets that it is finite on compact sets
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad On translation and dilation invariant Lebesgue measure: Folland's text
  • · Replies 2 ·
Replies
2
Views
3K
Elementary Sets and their Measures
  • · Replies 6 ·
Replies
6
Views
3K
Proof that algebraic numbers are countable
  • · Replies 6 ·
Replies
6
Views
3K