Is the Mandelbrot Set Lebesgue Measurable?

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Discussion Overview

The discussion centers around the Lebesgue measurability of the Mandelbrot set, exploring whether it can be classified as Lebesgue measurable and examining its measure and dimensional properties. The conversation includes technical aspects and conjectures related to the set's measure.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants question the Lebesgue measure of the Mandelbrot set and whether it is known.
  • One participant asserts that the Mandelbrot set is not Lebesgue measurable and suggests that the discussion may be conflating measurability with dimensionality.
  • A participant cites a Wikipedia estimate for the measure of the Mandelbrot set, noting uncertainty about its reliability, while also stating that two large areas of the set have positive measures.
  • Another participant challenges the claim of non-measurability by stating that every closed set is Lebesgue measurable and that the Mandelbrot set is closed.
  • There are repeated inquiries about the basis for claiming the set is not Lebesgue measurable, indicating a desire for clarification or evidence.

Areas of Agreement / Disagreement

Participants express conflicting views regarding the Lebesgue measurability of the Mandelbrot set, with no consensus reached on the matter.

Contextual Notes

There are references to the dimensional properties of the Mandelbrot set and the reliability of sources cited for its measure, indicating potential limitations in the discussion.

Dragonfall
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What is the Lebesgue measure of the Mandelbrot set?
 
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Is it known?
 
The mandelbrot set is not "Lebesque measurable". Is is possible that you are referring to the dimension of the set?
 
According to Wikipedia, the measure is estimated to be 1.506 591 77 ± 0.000 000 08, and it is conjectured to be exactly \sqrt{6\pi-1} - e

edit: But after reading the source... I'm really not sure if I would trust that too well. However, the two large areas of the Mandelbrot set each definitely have positive measures
 
Last edited:
How do you know that it's not Lebesgue measurable?
 
HallsofIvy said:
The mandelbrot set is not "Lebesque measurable". Is is possible that you are referring to the dimension of the set?

<< insult deleted by Mentors >> every closed set is Lebesgue measurable.

The Mandelbrot Set is closed.

J
 
Last edited by a moderator:
Dragonfall said:
How do you know that it's not Lebesgue measurable?

I know this is a very old post, but read what I just posted in reply.

J
 

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