uniqueness of RN derivatives up ae eq (#914)

Author: affeldt-aist

CHANGELOG_UNRELEASED.md

@@ -22,6 +22,14 @@
 - in file `topology.v`,
   + new definitions `basis`, and `second_countable`.
   + new lemmas `clopen_countable` and `compact_countable_base`.
+- in `classical_sets.v`:
+  + lemmas `set_eq_le`, `set_neq_lt`
+- in `set_interval.v`:
+  + lemma `set_lte_bigcup`
+- in `lebesgue_integral.v`:
+  + lemmas `emeasurable_fun_lt`, `emeasurable_fun_le`, `emeasurable_fun_eq`,
+    `emeasurable_fun_neq`
+  + lemma `integral_ae_eq`
 
 ### Changed
 

classical/classical_sets.v

@@ -1018,6 +1018,19 @@ Notation setvI := setICl.
 #[deprecated(since="mathcomp-analysis 0.6", note="Use setICr instead.")]
 Notation setIv := setICr.
 
+Section set_order.
+Import Order.TTheory.
+
+Lemma set_eq_le d (rT : porderType d) T (f g : T -> rT) :
+  [set x | f x = g x] = [set x | (f x <= g x)%O] `&` [set x | (f x >= g x)%O].
+Proof. by apply/seteqP; split => [x/= ->//|x /andP]; rewrite -eq_le =>/eqP. Qed.
+
+Lemma set_neq_lt d (rT : orderType d) T (f g : T -> rT) :
+  [set x | f x != g x ] = [set x | (f x < g x)%O] `|` [set x | (f x > g x)%O].
+Proof. by apply/seteqP; split => [x/=|x /=]; rewrite neq_lt => /orP. Qed.
+
+End set_order.
+
 Lemma image2E {TA TB rT : Type} (A : set TA) (B : set TB) (f : TA -> TB -> rT) :
   [set f x y | x in A & y in B] = uncurry f @` (A `*` B).
 Proof.

theories/lebesgue_integral.v

@@ -3,8 +3,8 @@ From HB Require Import structures.
 From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap.
 From mathcomp.classical Require Import boolp classical_sets functions.
 From mathcomp.classical Require Import cardinality fsbigop mathcomp_extra.
-Require Import signed reals ereal topology normedtype sequences esum measure.
-Require Import lebesgue_measure numfun.
+Require Import signed reals ereal topology normedtype sequences real_interval.
+Require Import esum measure lebesgue_measure numfun.
 
 (******************************************************************************)
 (*                            Lebesgue Integral                               *)
@@ -1858,6 +1858,42 @@ Proof. by move=> mf; exact/(emeasurable_funM _ mf)/measurable_fun_cst. Qed.
 
 End emeasurable_fun.
 
+Section measurable_fun_measurable2.
+Local Open Scope ereal_scope.
+Context d (T : measurableType d) (R : realType).
+Variables (D : set T) (mD : measurable D).
+Implicit Types f g : T -> \bar R.
+
+Lemma emeasurable_fun_lt f g : measurable_fun D f -> measurable_fun D g ->
+  measurable (D `&` [set x | f x < g x]).
+Proof.
+move=> mf mg; under eq_set do rewrite -sube_gt0.
+by apply: emeasurable_fun_o_infty => //; exact: emeasurable_funB.
+Qed.
+
+Lemma emeasurable_fun_le f g : measurable_fun D f -> measurable_fun D g ->
+  measurable (D `&` [set x | f x <= g x]).
+Proof.
+move=> mf mg; under eq_set do rewrite -sube_le0.
+by apply: emeasurable_fun_infty_c => //; exact: emeasurable_funB.
+Qed.
+
+Lemma emeasurable_fun_eq f g : measurable_fun D f -> measurable_fun D g ->
+  measurable (D `&` [set x | f x = g x]).
+Proof.
+move=> mf mg; rewrite set_eq_le setIIr.
+by apply: measurableI; apply: emeasurable_fun_le.
+Qed.
+
+Lemma emeasurable_fun_neq f g : measurable_fun D f -> measurable_fun D g ->
+  measurable (D `&` [set x | f x != g x]).
+Proof.
+move=> mf mg; rewrite set_neq_lt setIUr.
+by apply: measurableU; exact: emeasurable_fun_lt.
+Qed.
+
+End measurable_fun_measurable2.
+
 Section ge0_integral_sum.
 Local Open Scope ereal_scope.
 Context d (T : measurableType d) (R : realType).
@@ -4132,6 +4168,72 @@ Qed.
 
 End ae_ge0_le_integral.
 
+Section integral_ae_eq.
+Local Open Scope ereal_scope.
+Context d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}).
+
+Let integral_measure_lt (D : set T) (mD : measurable D) (g f : T -> \bar R) :
+  mu.-integrable D f -> mu.-integrable D g ->
+  (forall E, measurable E -> \int[mu]_(x in E) f x = \int[mu]_(x in E) g x) ->
+  mu (D `&` [set x | g x < f x]) = 0.
+Proof.
+move=> mf mg fg; pose E j := D `&` [set x | f x - g x >= j.+1%:R^-1%:E].
+have mE j : measurable (E j).
+  rewrite /E; apply: emeasurable_fun_le => //; first exact: measurable_fun_cst.
+  by apply/(emeasurable_funD mf.1)/emeasurable_funN; case: mg.
+have muE j : mu (E j) = 0.
+  apply/eqP; rewrite eq_le measure_ge0// andbT.
+  have fg0 : \int[mu]_(x in E j) (f \- g) x = 0.
+    rewrite integralB//; last 2 first.
+      by apply: integrableS mf => //; exact: subIsetl.
+      by apply: integrableS mg => //; exact: subIsetl.
+    rewrite fg// subee// fin_num_abs (le_lt_trans (le_abse_integral _ _ _))//.
+      by apply: measurable_funS mg.1 => //; first exact: subIsetl.
+    apply: le_lt_trans mg.2; apply: subset_integral => //; last exact: subIsetl.
+    exact: measurable_funT_comp mg.1.
+  suff : mu (E j) <= j.+1%:R%:E * \int[mu]_(x in E j) (f \- g) x.
+    by rewrite fg0 mule0.
+  apply: (@le_trans _ _ (j.+1%:R%:E * \int[mu]_(x in E j) j.+1%:R^-1%:E)).
+    by rewrite integral_cst// muleA -EFinM divrr ?unitfE// mul1e.
+  rewrite lee_pmul//; first exact: integral_ge0.
+  apply: ge0_le_integral => //; [exact: measurable_fun_cst| | |by move=> x []].
+  - by move=> x [_/=]; exact: le_trans.
+  - apply: emeasurable_funB.
+    + by apply: measurable_funS mf.1 => //; exact: subIsetl.
+    + by apply: measurable_funS mg.1 => //; exact: subIsetl.
+have nd_E : {homo E : n0 m / (n0 <= m)%N >-> (n0 <= m)%O}.
+  move=> i j ij; apply/subsetPset => x [Dx /= ifg]; split => //.
+  by move: ifg; apply: le_trans; rewrite lee_fin lef_pinv// ?posrE// ler_nat.
+rewrite set_lte_bigcup.
+have /cvg_lim h1 : mu \o E --> 0 by apply: cvg_near_cst; exact: nearW.
+have := @nondecreasing_cvg_mu _ _ _ mu E mE (bigcupT_measurable E mE) nd_E.
+by move/cvg_lim => h2; rewrite setI_bigcupr -h2// h1.
+Qed.
+
+Lemma integral_ae_eq (D : set T) (mD : measurable D) (g f : T -> \bar R) :
+  mu.-integrable D f -> mu.-integrable D g ->
+  (forall E, measurable E -> \int[mu]_(x in E) f x = \int[mu]_(x in E) g x) ->
+  ae_eq mu D f g.
+Proof.
+move=> mf mg fg.
+have mugf : mu (D `&` [set x | g x < f x]) = 0 by exact: integral_measure_lt.
+have mufg : mu (D `&` [set x | f x < g x]) = 0.
+  by apply: integral_measure_lt => // E mE; rewrite fg.
+have h : ~` [set x | D x -> f x = g x] = D `&` [set x | f x != g x].
+  apply/seteqP; split => [x/= /not_implyP[? /eqP]//|x/= [Dx fgx]].
+  by apply/not_implyP; split => //; exact/eqP.
+apply/negligibleP.
+  by rewrite h; apply: emeasurable_fun_neq => //; [case: mf|case: mg].
+rewrite h set_neq_lt setIUr measureU//.
+- by rewrite [X in X + _]mufg add0e [LHS]mugf.
+- by apply: emeasurable_fun_lt => //; [case: mf|case: mg].
+- by apply: emeasurable_fun_lt => //; [case: mg|case: mf].
+- apply/seteqP; split => [x [[Dx/= + [_]]]|//].
+  by move=> /lt_trans => /[apply]; rewrite ltxx.
+Qed.
+
+End integral_ae_eq.
+
 (******************************************************************************)
 (* * product measure                                                          *)
 (******************************************************************************)

theories/real_interval.v

@@ -275,7 +275,7 @@ Coercion ereal_of_itv_bound T (b : itv_bound T) : \bar T :=
   match b with BSide _ y => y%:E | +oo%O => +oo%E | -oo%O => -oo%E end.
 Arguments ereal_of_itv_bound T !b.
 
-Section erealDomainType.
+Section itv_realDomainType.
 Context (R : realDomainType).
 
 Lemma le_bnd_ereal (a b : itv_bound R) : (a <= b)%O -> (a <= b)%E.
@@ -325,7 +325,32 @@ rewrite set_itvE predeqE => x; split => /=.
 - by move: x => [x h|//|/(_ erefl)]; rewrite ?ltNyr.
 Qed.
 
-End erealDomainType.
+End itv_realDomainType.
+
+Section set_ereal.
+Context (R : realType) T (f g : T -> \bar R).
+Local Open Scope ereal_scope.
+
+Let E j := [set x | f x - g x >= j.+1%:R^-1%:E].
+
+Lemma set_lte_bigcup : [set x | f x > g x] = \bigcup_j E j.
+Proof.
+apply/seteqP; split => [x/=|x [n _]]; last first.
+  by rewrite /E/= -sube_gt0; apply: lt_le_trans.
+move gxE : (g x) => gx; case: gx gxE => [gx| |gxoo fxoo]; last 2 first.
+  - by case: (f x).
+  - by exists 0%N => //; rewrite /E/= gxoo addey// ?leey// -ltNye.
+move fxE : (f x) => fx; case: fx fxE => [fx fxE gxE|fxoo gxE _|//]; last first.
+  by exists 0%N => //; rewrite /E/= fxoo gxE// addye// leey.
+rewrite lte_fin -subr_gt0 => fgx; exists `|floor (fx - gx)^-1%R|%N => //.
+rewrite /E/= -natr1 natr_absz ger0_norm ?floor_ge0 ?invr_ge0; last exact/ltW.
+rewrite fxE gxE lee_fin -[leRHS]invrK lef_pinv//.
+- by apply/ltW; rewrite lt_succ_floor.
+- by rewrite posrE// ltr_spaddr// ler0z floor_ge0 invr_ge0 ltW.
+- by rewrite posrE invr_gt0.
+Qed.
+
+End set_ereal.
 
 Lemma disj_itv_Rhull {R : realType} (A B : set R) : A `&` B = set0 ->
   is_interval A -> is_interval B -> disjoint_itv (Rhull A) (Rhull B).