Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

lemmas about Lebesgue's outer measure #1293

Merged
merged 2 commits into from
Aug 20, 2024

Conversation

affeldt-aist
Copy link
Member

Motivation for this change

Four lemmas about the Lebesgue measure.
The second one is actually the most interesting because we use to prove
an important corollary of the Vitali's theorem (PR to come)

  • outer_measure_open_itv_cover:
    definition of the outer measure in terms of covers of open intervals
    (proposition 20 in Li)
  • outer_measure_open:
    for a set A with bounded outer measure and an epsilon,
    there is an open that contains A and whose measure is
    less than the outer measure A + epsilon,
    (proposition 21 1) in Li)
  • outer_measure_Gdelta:
    for a set A with bounded outer measure, there exists a G-delta sets
    that contains A and whose measure is the same as A
    (proposition 21 2) in Li)
  • negligible_outer_measure:
    a set N is negligible for the lebesgue measure iff its outer measure is 0
    (corollary 22 in Li)
Checklist
  • added corresponding entries in CHANGELOG_UNRELEASED.md
  • added corresponding documentation in the headers

Reference: How to document

Reminder to reviewers

@affeldt-aist affeldt-aist added the enhancement ✨ This issue/PR is about adding new features enhancing the library label Aug 13, 2024
@affeldt-aist affeldt-aist requested a review from zstone1 August 13, 2024 06:25
@affeldt-aist affeldt-aist added this to the 1.4.0 milestone Aug 13, 2024
Copy link
Contributor

@zstone1 zstone1 left a comment

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Look pretty good, just one comment on a slight generalization.


Definition mu := (@lebesgue_measure R).

Lemma outer_measure_open A : (l^* A)%mu < +oo ->
Copy link
Contributor

@zstone1 zstone1 Aug 13, 2024

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Any reason not to phrase it as

(l^*)%mu A = ereal_inf [set (l^* A)%mu | open U /\ A `<=` U]

so it is more compatible with outer_measure_open_itv_cover's phrasing? It also has the benefit of being true in the unbounded case.

Copy link
Member Author

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

(l^*)%mu U = ereal_inf [set (l^* A)%mu | open U /\ A `<=` U] (note the typo fix)
has been added as a lemma to the last commit but I am not sure how to exploit it

in any case, it suggested that outer_measure_Gdelta and outer_measure_open_le (new name) do not need the boundedness hypothesis so it has been removed

@affeldt-aist affeldt-aist force-pushed the lebesgue_measure_20240813 branch from 0fffc6b to 51d8f98 Compare August 18, 2024 04:06
@affeldt-aist affeldt-aist force-pushed the lebesgue_measure_20240813 branch from 51d8f98 to 21f96a1 Compare August 18, 2024 04:32
@affeldt-aist affeldt-aist merged commit 943b812 into math-comp:master Aug 20, 2024
22 checks passed
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment
Labels
enhancement ✨ This issue/PR is about adding new features enhancing the library
Projects
None yet
Development

Successfully merging this pull request may close these issues.

2 participants