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.

Sign up for GitHub

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

deprecate approximation and make its interface accessible #1421

Merged
merged 1 commit into from
Dec 3, 2024
Merged

deprecate approximation and make its interface accessible #1421

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
24 changes: 24 additions & 0 deletions CHANGELOG_UNRELEASED.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -12,17 +12,41 @@

- in `mathcomp_extra.v`:
+ lemma `partition_disjoint_bigfcup`
- in `lebesgue_measure.v`:
+ lemma `measurable_indicP`

- in `lebesgue_integral.v`:
+ definition `dyadic_approx` (was `Let A`)
+ definition `integer_approx` (was `Let B`)
+ lemma `measurable_sum`
+ lemma `integrable_indic`

### Changed

- in `lebesgue_integrale.v`
+ change implicits of `measurable_funP`

### Renamed

- in `lebesgue_measure.v`:
+ `measurable_fun_indic` -> `measurable_indic`
+ `emeasurable_fun_sum` -> `emeasurable_sum`
+ `emeasurable_fun_fsum` -> `emeasurable_fsum`
+ `ge0_emeasurable_fun_sum` -> `ge0_emeasurable_sum`

### Generalized

### Deprecated

- in file `lebesgue_integral.v`:
+ lemma `approximation`

### Removed

- in `lebesgue_integral.v`:
+ lemma `measurable_indic` (was uselessly specializing `measurable_fun_indic` (now `measurable_indic`) from `lebesgue_measure.v`)
+ notation `measurable_fun_indic` (deprecation since 0.6.3)

### Infrastructure

### Misc
22 changes: 10 additions & 12 deletions theories/charge.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1837,7 +1837,7 @@ have int_f_nuT : \int[mu]_x f x = nu setT.
by apply: eq_eseriesr => i _; rewrite int_f_E// setTI.
rewrite -UET measure_bigcup//.
by apply: eq_eseriesl => // x; rewrite in_setT.
have mf : measurable_fun setT f by exact: ge0_emeasurable_fun_sum.
have mf : measurable_fun setT f by exact: ge0_emeasurable_sum.
have fi : mu.-integrable setT f.
apply/integrableP; split => //.
under eq_integral do (rewrite gee0_abs; last exact: nneseries_ge0).
Expand Down Expand Up @@ -1921,29 +1921,29 @@ Lemma change_of_variables f E : (forall x, 0 <= f x) ->
\int[mu]_(x in E) (f x * ('d nu '/d mu) x) = \int[nu]_(x in E) f x.
Proof.
move=> f0 mE mf; set g := 'd nu '/d mu.
have [h [ndh hf]] := approximation mE mf (fun x _ => f0 x).
pose h := nnsfun_approx mE mf.
have -> : \int[nu]_(x in E) f x =
lim (\int[nu]_(x in E) (EFin \o h n) x @[n --> \oo]).
have fE x : E x -> f x = lim ((EFin \o h n) x @[n --> \oo]).
by move=> Ex; apply/esym/cvg_lim => //; exact: hf.
by move=> Ex; apply/esym/cvg_lim => //; exact: cvg_nnsfun_approx.
under eq_integral => x /[!inE] /fE -> //.
apply: monotone_convergence => //.
- move=> n; apply/measurable_EFinP.
by apply: (measurable_funS measurableT) => //; exact/measurable_funP.
- by move=> n x Ex //=; rewrite lee_fin.
- by move=> x Ex a b /ndh /=; rewrite lee_fin => /lefP.
- by move=> x Ex a b ab; rewrite lee_fin; exact/lefP/nd_nnsfun_approx.
have -> : \int[mu]_(x in E) (f \* g) x =
lim (\int[mu]_(x in E) ((EFin \o h n) \* g) x @[n --> \oo]).
have fg x :E x -> f x * g x = lim (((EFin \o h n) \* g) x @[n --> \oo]).
by move=> Ex; apply/esym/cvg_lim => //; apply: cvgeMr;
[exact: f_fin_num|exact: hf].
[exact: f_fin_num|exact: cvg_nnsfun_approx].
under eq_integral => x /[!inE] /fg -> //.
apply: monotone_convergence => [//| | |].
- move=> n; apply/emeasurable_funM; apply/measurable_funTS.
exact/measurable_EFinP.
exact: measurable_int (f_integrable _).
- by move=> n x Ex //=; rewrite mule_ge0 ?lee_fin//=; exact: f_ge0.
- by move=> x Ex a b /ndh /= /lefP hahb; rewrite lee_wpmul2r ?lee_fin// f_ge0.
- by move=> x Ex a b ab/=; rewrite lee_wpmul2r ?lee_fin ?f_ge0//; exact/lefP/nd_nnsfun_approx.
suff suf n : \int[mu]_(x in E) ((EFin \o h n) x * g x) =
\int[nu]_(x in E) (EFin \o h n) x.
by under eq_fun do rewrite suf.
Expand Down Expand Up @@ -2005,18 +2005,16 @@ Local Notation "'d nu '/d mu" := (f nu mu).
Lemma chain_rule E : nu `<< mu -> mu `<< la -> measurable E ->
ae_eq la E ('d nu '/d la) ('d nu '/d mu \* 'd mu '/d la).
Proof.
move=> numu mula mE; have nula := measure_dominates_trans numu mula.
move=> numu mula mE.
have mf : measurable_fun E ('d nu '/d mu).
exact/measurable_funTS/(measurable_int _ (f_integrable _)).
have [h [ndh hf]] := approximation mE mf (fun x _ => f_ge0 numu x).
apply: integral_ae_eq => //.
- apply: (integrableS measurableT) => //.
apply: f_integrable.
exact: (measure_dominates_trans numu mula).
- apply: (integrableS measurableT) => //; apply: f_integrable.
exact: measure_dominates_trans numu mula.
- apply: emeasurable_funM => //.
exact/measurable_funTS/(measurable_int _ (f_integrable _)).
- move=> A AE mA; rewrite change_of_variables//.
+ by rewrite -!f_integral.
+ by rewrite -!f_integral//; exact: measure_dominates_trans numu mula.
+ exact: f_ge0.
+ exact: measurable_funS mf.
Qed.
Expand Down
54 changes: 31 additions & 23 deletions theories/kernel.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -131,8 +131,7 @@ Definition kseries : X -> {measure set Y -> \bar R} :=
Lemma measurable_fun_kseries (U : set Y) :
measurable U -> measurable_fun [set: X] (kseries ^~ U).
Proof.
move=> mU.
by apply: ge0_emeasurable_fun_sum => // n _; exact/measurable_kernel.
by move=> mU; apply: ge0_emeasurable_sum => // n _; exact/measurable_kernel.
Qed.

HB.instance Definition _ :=
Expand Down Expand Up @@ -546,7 +545,7 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => \sum_(r \in range (k_ n))
apply/measurable_EFinP/measurableT_comp => [//|].
exact/measurableT_comp.
- by move=> m y _; rewrite nnfun_muleindic_ge0.
apply: emeasurable_fun_fsum => // r.
apply: emeasurable_fsum => // r.
rewrite [X in measurable_fun _ X](_ : _ = fun x => r%:E *
\int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E); last first.
apply/funext => x; under eq_integral do rewrite EFinM.
Expand All @@ -571,24 +570,30 @@ Lemma measurable_fun_integral_finite_kernel (l : R.-fker X ~> Y)
(k0 : forall z, 0 <= k z) (mk : measurable_fun [set: X * Y] k) :
measurable_fun [set: X] (fun x => \int[l x]_y k (x, y)).
Proof.
have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x).
apply: (measurable_fun_xsection_integral ndk_ (k_k ^~ Logic.I)) => n r.
have [l_ hl_] := measure_uub l.
by apply: measurable_fun_xsection_finite_kernel => // /[!inE].
pose k_ := nnsfun_approx measurableT mk.
apply: (@measurable_fun_xsection_integral _ k_).
- by move=> a b ab; exact/nd_nnsfun_approx.
- by move=> z; exact/cvg_nnsfun_approx.
- move => n r.
have [l_ hl_] := measure_uub l.
by apply: measurable_fun_xsection_finite_kernel => // /[!inE].
Qed.

Lemma measurable_fun_integral_sfinite_kernel (l : R.-sfker X ~> Y)
(k0 : forall t, 0 <= k t) (mk : measurable_fun [set: X * Y] k) :
measurable_fun [set: X] (fun x => \int[l x]_y k (x, y)).
Proof.
have [k_ [ndk_ k_k]] := approximation measurableT mk (fun xy _ => k0 xy).
apply: (measurable_fun_xsection_integral ndk_ (k_k ^~ Logic.I)) => n r.
have [l_ hl_] := sfinite_kernel l.
rewrite (_ : (fun x => _) = (fun x =>
mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x))); last first.
by apply/funext => x; rewrite hl_//; exact/measurable_xsection.
apply: ge0_emeasurable_fun_sum => // m _.
by apply: measurable_fun_xsection_finite_kernel => // /[!inE].
pose k_ := nnsfun_approx measurableT mk.
apply: (@measurable_fun_xsection_integral _ k_).
- by move=> a b ab; exact/nd_nnsfun_approx.
- by move=> z; exact/cvg_nnsfun_approx.
- move => n r.
have [l_ hl_] := sfinite_kernel l.
rewrite (_ : (fun x => _) = (fun x =>
mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x))); last first.
by apply/funext => x; rewrite hl_//; exact/measurable_xsection.
apply: ge0_emeasurable_sum => // m _.
by apply: measurable_fun_xsection_finite_kernel => // /[!inE].
Qed.

End measurable_fun_integral_finite_sfinite.
Expand Down Expand Up @@ -1007,8 +1012,11 @@ Lemma measurable_fun_integral_kernel
(k : Y -> \bar R) (k0 : forall z, 0 <= k z) (mk : measurable_fun [set: Y] k) :
measurable_fun [set: X] (fun x => \int[l x]_y k y).
Proof.
have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x).
by apply: (measurable_fun_preimage_integral ndk_ k_k) => n r; exact/ml.
pose k_ := nnsfun_approx measurableT mk.
apply: (@measurable_fun_preimage_integral _ _ _ _ _ _ k_) => //.
- by move=> a b ab; exact/nd_nnsfun_approx.
- by move=> z _; exact/cvg_nnsfun_approx.
- by move=> n r; exact/ml.
Qed.

End measurable_fun_integral_kernel.
Expand Down Expand Up @@ -1077,13 +1085,13 @@ Lemma integral_kcomp x f : (forall z, 0 <= f z) -> measurable_fun [set: Z] f ->
\int[kcomp l k x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z).
Proof.
move=> f0 mf.
have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z).
pose f_ := nnsfun_approx measurableT mf.
transitivity (\int[kcomp l k x]_z (lim ((f_ n z)%:E @[n --> \oo]))).
by apply/eq_integral => z _; apply/esym/cvg_lim => //=; exact: f_f.
by apply/eq_integral => z _; apply/esym/cvg_lim => //=; exact: cvg_nnsfun_approx.
rewrite monotone_convergence//; last 3 first.
by move=> n; exact/measurable_EFinP.
by move=> n z _; rewrite lee_fin.
by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin.
by move=> z _ a b ab; rewrite lee_fin; exact/lefP/nd_nnsfun_approx.
rewrite (_ : (fun _ => _) =
(fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first.
by apply/funext => n; rewrite integral_kcomp_nnsfun.
Expand All @@ -1099,12 +1107,12 @@ transitivity (\int[l x]_y lim ((\int[k (x, y)]_z (f_ n z)%:E) @[n --> \oo])).
+ exact/measurable_EFinP.
+ by move=> z _; rewrite lee_fin.
+ exact/measurable_EFinP.
+ by move: ab => /ndf_ /lefP ab z _; rewrite lee_fin.
+ by move=> z _; rewrite lee_fin; exact/lefP/nd_nnsfun_approx.
apply: eq_integral => y _; rewrite -monotone_convergence//; last 3 first.
- by move=> n; exact/measurable_EFinP.
- by move=> n z _; rewrite lee_fin.
- by move=> z _ a b /ndf_ /lefP; rewrite lee_fin.
by apply: eq_integral => z _; apply/cvg_lim => //; exact: f_f.
- by move=> z _ a b ab; rewrite lee_fin; exact/lefP/nd_nnsfun_approx.
by apply: eq_integral => z _; apply/cvg_lim => //; exact: cvg_nnsfun_approx.
Qed.

End integral_kcomp.
Loading
Loading