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[Merged by Bors] - feat: generalize some lemmas using withDensity_apply' #8383

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[Merged by Bors] - feat: generalize some lemmas using withDensity_apply' #8383

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@RemyDegenne RemyDegenne commented Nov 13, 2023 •
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@sgouezel added a version of withDensity_apply that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about rnDeriv, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.

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@RemyDegenne RemyDegenne added awaiting-review The author would like community review of the PR t-measure-probability Measure theory / Probability theory labels Nov 13, 2023
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RemyDegenne commented Nov 13, 2023 •
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Perhaps withDensity_apply' could also hold for countable sums of (sigma)-finite measures? That may require removing the MeasurableSet assumption in sum_apply (under some assumptions).

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Do we have a property IsSFiniteMeasure (of which sigma-finite measures would be a special case)? The lemma withDensity_apply should still be true under this weaker assumption, with the same proof.

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I've defined s-finite measures in #8405, and proved there that withDensity_apply' works for s-finite measures. Would it help for the applications you have in mind?

sgouezel and others added 2 commits November 14, 2023 10:18
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@RemyDegenne RemyDegenne added WIP Work in progress and removed awaiting-review The author would like community review of the PR labels Nov 14, 2023
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Thanks to #8405 I could remove the MeasurableSet assumption from kernel.withDensity_apply', since values of s-finite kernels are s-finite measures.

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RemyDegenne commented Nov 14, 2023
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Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
rw [set_integral_toReal_rnDeriv_eq_withDensity' hs, Measure.withDensity_rnDeriv_eq _ _ hμν]
#align measure_theory.measure.with_density_rn_deriv_to_real_eq MeasureTheory.Measure.set_integral_toReal_rnDeriv'

lemma set_integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
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I did not replace SigmaFinite ν by SFinite ν here because then HaveLebesgueDecomposition μ ν cannot be inferred, which leads to this question: can the Lebesgue decomposition theorem be generalized to s-finite measures?
(if yes, let's not do it in this PR)

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Yes, many things hold for s-finite measures instead of sigma-finite (Fubini, for instance!). I'm not sure it's worth doing the change unless there are applications where it is useful to weaken some assumptions.

@RemyDegenne RemyDegenne added awaiting-review The author would like community review of the PR and removed WIP Work in progress labels Nov 14, 2023
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bors r+
Thanks!

@github-actions github-actions bot added ready-to-merge This PR has been sent to bors. and removed awaiting-review The author would like community review of the PR labels Nov 14, 2023
mathlib-bors bot pushed a commit that referenced this pull request Nov 14, 2023
@RemyDegenne @sgouezel
feat: generalize some lemmas using withDensity_apply' (#8383)
21b5939 
@sgouezel added a version of `withDensity_apply` that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about `rnDeriv`, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.



Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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mathlib-bors bot commented Nov 14, 2023

Pull request successfully merged into master.

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@mathlib-bors mathlib-bors bot changed the title feat: generalize some lemmas using withDensity_apply' [Merged by Bors] - feat: generalize some lemmas using withDensity_apply' Nov 14, 2023
@mathlib-bors mathlib-bors bot closed this Nov 14, 2023
@mathlib-bors mathlib-bors bot deleted the RD_withDensity branch November 14, 2023 13:04
alexkeizer pushed a commit that referenced this pull request Nov 17, 2023
@RemyDegenne @sgouezel @alexkeizer
feat: generalize some lemmas using withDensity_apply' (#8383)
8c02958 
@sgouezel added a version of `withDensity_apply` that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about `rnDeriv`, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.



Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
alexkeizer pushed a commit that referenced this pull request Nov 21, 2023
@RemyDegenne @sgouezel @alexkeizer
feat: generalize some lemmas using withDensity_apply' (#8383)
97bc4a7 
@sgouezel added a version of `withDensity_apply` that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about `rnDeriv`, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.



Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
grunweg pushed a commit that referenced this pull request Dec 15, 2023
@RemyDegenne @sgouezel @grunweg
feat: generalize some lemmas using withDensity_apply' (#8383)
fc29397 
@sgouezel added a version of `withDensity_apply` that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about `rnDeriv`, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.



Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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