lemmas about Lebesgue's outer measure#1293
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affeldt-aist merged 2 commits intomath-comp:masterfrom Aug 20, 2024
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Look pretty good, just one comment on a slight generalization.
theories/lebesgue_measure.v
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| Definition mu := (@lebesgue_measure R). | ||
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| Lemma outer_measure_open A : (l^* A)%mu < +oo -> |
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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.
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(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
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zstone1
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Aug 19, 2024
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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
CHANGELOG_UNRELEASED.mdReference: How to document
Reminder to reviewers