How to prove that cuboids are Lebesgue measurable?

  • Context: Graduate 
  • Thread starter Thread starter Maxi1995
  • Start date Start date
  • Tags Tags
    Measurable
Click For Summary

Discussion Overview

The discussion centers on the proof of the Lebesgue measurability of cuboids, exploring the foundational aspects of measure theory, particularly in relation to intervals and their three-dimensional analogs. Participants consider various approaches and implications of the measure definition.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant suggests starting with the definition of the measure of an interval as its length, noting that a cuboid is the three-dimensional analog of this concept.
  • Another participant proposes that one could define the measure of a cuboid as its volume or consider three-dimensional space as the direct product of three lines.
  • Concerns are raised about the implications of rotating a cuboid, with one participant indicating that the measure's invariance under rotations needs to be proven.
  • Some participants mention that the coordinate system can be rotated along with the cuboid, but this is noted as separate from the construction of the measure itself.
  • A suggestion is made to search for "Lebesgue measure lecture notes" for additional resources on the topic.

Areas of Agreement / Disagreement

Participants express differing views on the implications of rotation and the necessity of proving invariance under such transformations. The discussion remains unresolved regarding the specific proof steps and the treatment of rotations in the context of Lebesgue measure.

Contextual Notes

There are limitations regarding the assumptions made about the definition of measure and the treatment of rotations, which are not fully explored in the discussion.

Maxi1995
Messages
14
Reaction score
0
Hello,
how do I have to start to prove that cuboids are measurable in the context of the Lebesgue measure?
Best wishes
Maxi
 
Last edited:
Physics news on Phys.org
Measure theory starts from the definition of measure of an interval is equal to its length. A cuboid is just the 3d analog of an interval. You can start with measure equals volume or you can define 3 space as the direct product of 3 lines and go on from there.
 
mathman said:
Measure theory starts from the definition of measure of an interval is equal to its length. A cuboid is just the 3d analog of an interval. You can start with measure equals volume or you can define 3 space as the direct product of 3 lines and go on from there.
I think the subtlety comes when the cuboid is rotated.
 
The coordinate system can be rotated with it.
 
mathman said:
The coordinate system can be rotated with it.
Yes, but that is not part of the construction of the measure. One still needs to prove that it is invariant under rotations.
 
Gooogle "Lebesgue measure lecture notes" - this will get you more answers.
 
Thank you for yor answers, I'm going to think about it and give a sign in case of further needed help. :)
 

Similar threads

Undergrad On commutativity of convolution
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Opinion about an introduction of the Lebesgue integral in "Introduction to Hilbert spaces with applications"
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Compute Lebesgue integral as (improper) Riemann integral
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Lebesgue Outer Measure .... Carothers, Proposition 16.2 (i) ....
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad On translation and dilation invariant Lebesgue measure: Folland's text
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Exercise 2.23 in Folland's real analysis text
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Parallel rectangles contained in oblique rectangle
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Basic measurement theory question
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad On theorem 1.19 in Folland's and completion of measure
  • · Replies 1 ·
Replies
1
Views
3K