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affeldt-aist merged 1 commit into from
Mar 26, 2024
Merged

subset_integral naming #1194

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1 change: 1 addition & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -121,6 +121,7 @@

- in `lebesgue_integral.v`
+ `integral_setU` -> `ge0_integral_setU`
+ `subset_integral` -> `ge0_subset_integral`

### Generalized

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2 changes: 1 addition & 1 deletion theories/charge.v
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Expand Up @@ -1386,7 +1386,7 @@ under [in leRHS]eq_integral.
move=> x _; rewrite gee0_abs; last first.
exact: fRN_ge0.
over.
apply: subset_integral => //; first exact: measurable_fun_fRN.
apply: ge0_subset_integral => //; first exact: measurable_fun_fRN.
by move=> x _; exact: fRN_ge0.
Qed.

Expand Down
41 changes: 25 additions & 16 deletions theories/lebesgue_integral.v
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Expand Up @@ -30,6 +30,12 @@ Require Import esum measure lebesgue_measure numfun realfun function_spaces.
(* Main reference: *)
(* - Daniel Li, Intégration et applications, 2016 *)
(* *)
(* About the local naming convention: Lemmas about the Lebesgue integral *)
(* often appears in two flavors, depending on whether they are about non- *)
(* negative functions or about integrable functions. Lemmas about non- *)
(* negative functions are prefixed with ge0_, lemmas about integrable *)
(* functions are not. *)
(* *)
(* Detailed contents: *)
(* ```` *)
(* {mfun T >-> R} == type of real-valued measurable functions *)
Expand Down Expand Up @@ -2920,7 +2926,7 @@ rewrite ge0_integralD//; last 5 first.
by rewrite -integral_mkcondl setIC setUK -integral_mkcondl setKU.
Qed.

Lemma subset_integral (A B : set T) (mA : measurable A) (mB : measurable B)
Lemma ge0_subset_integral (A B : set T) (mA : measurable A) (mB : measurable B)
(f : T -> \bar R) : measurable_fun B f -> (forall x, B x -> 0 <= f x) ->
A `<=` B -> \int[mu]_(x in A) f x <= \int[mu]_(x in B) f x.
Proof.
Expand Down Expand Up @@ -2969,7 +2975,7 @@ apply: (@le_trans _ _ (\int[mu]_(x in D `&` [set x | `|g x| >= a%:E]) `|g x|)).
- by move=> x _ /=; exact/ltW.
- by apply: measurableT_comp => //; exact: measurable_funS mg.
- by move=> x /= [].
by apply: subset_integral => //; exact: measurableT_comp.
by apply: ge0_subset_integral => //; exact: measurableT_comp.
Qed.

End subset_integral.
Expand Down Expand Up @@ -3310,7 +3316,7 @@ Lemma integrableS (E D : set T) (f : T -> \bar R) :
Proof.
move=> mE mD DE /integrableP[mf ifoo]; apply/integrableP; split.
exact: measurable_funS mf.
apply: le_lt_trans ifoo; apply: subset_integral => //.
apply: le_lt_trans ifoo; apply: ge0_subset_integral => //.
exact: measurableT_comp.
Qed.

Expand Down Expand Up @@ -3748,7 +3754,8 @@ have h1 : mu.-integrable D f <-> mu.-integrable D (fun x => f x * (oneCN x)%:E).
rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first.
by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E) // lee_fin.
rewrite -integral_setI_indic//; case: intf => _; apply: le_lt_trans.
by apply: subset_integral => //; [exact:measurableI|exact:measurableT_comp].
by apply: ge0_subset_integral => //; [exact:measurableI|
exact:measurableT_comp].
apply/integrableP; split => //; rewrite (funID mN f) -/oneCN -/oneN.
have ? : measurable_fun D (fun x : T => f x * (oneCN x)%:E).
by apply: emeasurable_funM=> //; exact/EFin_measurable_fun/measurable_funTS.
Expand All @@ -3773,7 +3780,7 @@ have h2 : mu.-integrable (D `\` N) f <->
rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first.
by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E)// lee_fin.
rewrite -integral_setI_indic //; case: intCf => _; apply: le_lt_trans.
apply: subset_integral => //; [exact: measurableI|exact: measurableD|].
apply: ge0_subset_integral => //; [exact: measurableI|exact: measurableD|].
by apply: measurableT_comp => //; apply: measurable_funS mf => // ? [].
split.
move=> mDN A mA; rewrite setDE (setIC D) -setIA; apply: measurableI => //.
Expand Down Expand Up @@ -3874,7 +3881,7 @@ apply: (@le_trans _ _ (\int[mu]_(x in D `&` [set x | `|g x| >= a%:E]) f `|g x|))
exact: measurable_funS mg.
- by move=> x /= [Dx]; apply: f_nd;
rewrite inE /= in_itv /= andbT// lee_fin ltW.
apply: subset_integral => //; last by move=> x _ /=; rewrite f0.
apply: ge0_subset_integral => //; last by move=> x _ /=; rewrite f0.
by apply: measurableT_comp => //; exact: measurableT_comp.
Qed.

Expand Down Expand Up @@ -4454,8 +4461,8 @@ have muE j : mu (E j) = 0.
rewrite fg//; last apply: subIsetl.
rewrite subee// fin_num_abs (le_lt_trans (le_abse_integral _ _ _))//.
by apply: measurable_funS msg => //; first exact: subIsetl.
apply: le_lt_trans (integrableP _ _ _ itg).2; apply: subset_integral => //.
exact: measurableT_comp msg.
apply: le_lt_trans (integrableP _ _ _ itg).2.
apply: ge0_subset_integral => //; first exact: measurableT_comp msg.
exact: subIsetl.
suff : mu (E j) <= j.+1%:R%:E * \int[mu]_(x in E j) (f \- g) x.
by rewrite fg0 mule0.
Expand Down Expand Up @@ -5798,7 +5805,7 @@ Lemma integrable_locally D f : open D ->
mu.-integrable D (EFin \o f) -> locally_integrable D f.
Proof.
move=> oD /integrableP[mf foo]; split => //; first exact/EFin_measurable_fun.
move=> K KD cK; rewrite (le_lt_trans _ foo)// subset_integral//=.
move=> K KD cK; rewrite (le_lt_trans _ foo)// ge0_subset_integral//=.
- exact: compact_measurable.
- exact: open_measurable.
- apply/EFin_measurable_fun; apply: measurableT_comp => //.
Expand Down Expand Up @@ -5929,7 +5936,7 @@ Proof.
move=> [mf _ locf]; have [r0|r0] := leP r 0%R.
by rewrite (ball0 _ _).2// integral_set0.
rewrite (le_lt_trans _ (locf (closed_ball x r) _ (closed_ballR_compact _)))//.
apply: subset_integral => //; first exact: measurable_ball.
apply: ge0_subset_integral => //; first exact: measurable_ball.
- by apply: measurable_closed_ball; exact/ltW.
- apply: measurable_funTS; apply/measurableT_comp => //=.
exact: measurableT_comp.
Expand Down Expand Up @@ -5964,7 +5971,8 @@ move: a0; rewrite le_eqVlt => /predU1P[a0|a0].
move=> /= z; rewrite /ball/= => xzr; rewrite -(subrK x y) -(addrA (y - x)%R).
by rewrite (le_lt_trans (ler_normD _ _))// addrC ltrD// distrC.
have ? : k <= \int[mu]_(y in ball y (r + d)) `|(f y)%:E|.
apply: subset_integral =>//; [exact:measurable_ball|exact:measurable_ball|].
apply: ge0_subset_integral =>//; [exact:measurable_ball|
exact:measurable_ball|].
apply: measurable_funTS; apply: measurableT_comp => //=.
by apply/measurableT_comp => //=; case: locf.
have : iavg f (ball y (r + d)) <= HL f y.
Expand Down Expand Up @@ -6001,7 +6009,8 @@ have ? : ball x r `<=` ball y (r + d).
move=> /= z; rewrite /ball/= => xzr; rewrite -(subrK x y) -(addrA (y - x)%R).
by rewrite (le_lt_trans (ler_normD _ _))// addrC ltrD// distrC.
have ? : k <= \int[mu]_(z in ball y (r + d)) `|(f z)%:E|.
apply: subset_integral => //; [exact: measurable_ball|exact: measurable_ball|].
apply: ge0_subset_integral => //; [exact: measurable_ball|
exact: measurable_ball|].
by apply: measurable_funTS; do 2 apply: measurableT_comp => //.
have afxrdi : a%:E < (fine (mu (ball x (r + d))))^-1%:E *
\int[mu]_(z in ball y (r + d)) `|(f z)%:E|.
Expand Down Expand Up @@ -6080,7 +6089,7 @@ apply: (@le_trans _ _
rewrite -ge0_sume_distrr//; last by move=> x _; rewrite integral_ge0.
rewrite lee_wpmul2l//; first by rewrite lee_fin invr_ge0 ltW.
rewrite -ge0_integral_bigsetU//=.
- apply: subset_integral => //.
- apply: ge0_subset_integral => //.
+ by apply: bigsetU_measurable => ? ?; exact: measurable_ball.
+ by apply: measurableT_comp => //; apply: measurableT_comp => //; case: locf.
- by move=> n; exact: measurable_ball.
Expand Down Expand Up @@ -6479,7 +6488,7 @@ have locf_g_B n : locally_integrable setT (f_ k \- g_B n)%R.
+ by move/EFin_measurable_fun : mf.
+ exact: openT.
+ move=> K _ cK; rewrite (le_lt_trans _ intf)//=.
apply: subset_integral => //.
apply: ge0_subset_integral => //.
* exact: compact_measurable.
* by do 2 apply: measurableT_comp => //; move/EFin_measurable_fun : mf.
have mEHL i : measurable (HLf_g_Be i).
Expand Down Expand Up @@ -6791,7 +6800,7 @@ apply: (@le_trans _ _ ((fine (mu (E x n)))^-1%:E *
by apply/EFin_measurable_fun; case: locf => + _ _; exact: measurable_funS.
rewrite (@le_lt_trans _ _
(\int[mu]_(y in closed_ball x (r_ x n)%:num) `|(f y)%:E|))//.
apply: subset_integral => //.
apply: ge0_subset_integral => //.
+ exact: (hE _).1.
+ exact: measurable_closed_ball.
+ apply: measurableT_comp => //; apply/EFin_measurable_fun => //.
Expand All @@ -6816,7 +6825,7 @@ rewrite muleA lee_pmul//.
* apply: le_abse_integral => //; first exact: (hE x).1.
apply/EFin_measurable_fun; apply/measurable_funB => //.
by case: locf => + _ _; exact: measurable_funS.
* apply: subset_integral => //; first exact: (hE x).1.
* apply: ge0_subset_integral => //; first exact: (hE x).1.
exact: measurable_ball.
* apply/EFin_measurable_fun; apply: measurableT_comp => //.
apply/measurable_funB => //.
Expand Down