generalize SimpleFun codomain from realType to sigmaRingType

CHANGELOG_UNRELEASED.md

@@ -15,6 +15,14 @@
   + lemmas `zerofctE`, `onefctE`
 
 ### Added
+- new file `measurable_topology.v`:
+  + definition `borelType`
+  + lemmas `singleton_bigcap`, `measurable1`
+
+- in `simple_functions.v`:
+  + lemmas `mem_sfun_comp_pair`, `sfun_submod_closed`
+  + `{sfun aT >-> rT}` is an `lmodType` when `rT` is a `normedModType`
+
 - in `set_interval.v`:
   + lemmas `setU_itvob1`, `setU_1itvob`
 
@@ -393,6 +401,8 @@
 
 - in `simple_functions.v`:
   + lemmas `fctD`, `fctN`, `fctM`, `fctZ`
+  + structure `SimpleFun` (and notation `{sfun aT >-> _}`): codomain
+    generalized from `realType` to `sigmaRingType d'`, adding a display parameter `d'`;
 
 - in `ereal.v`:
   + lemmas `ge0_mule_fsumr`, `ge0_mule_fsuml`

_CoqProject

@@ -99,6 +99,7 @@ theories/derive.v
 theories/numfun.v
 
 theories/measure_theory/measurable_structure.v
+theories/measure_theory/measurable_topology.v
 theories/measure_theory/measure_function.v
 theories/measure_theory/counting_measure.v
 theories/measure_theory/dirac_measure.v

theories/lebesgue_integral_theory/simple_functions.v

@@ -6,6 +6,7 @@ From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
 From mathcomp Require Import cardinality reals fsbigop ereal topology tvs.
 From mathcomp Require Import normedtype sequences real_interval esum measure.
 From mathcomp Require Import lebesgue_measure numfun realfun measurable_realfun.
+From mathcomp Require Import measurable_function measurable_topology.
 
 (**md**************************************************************************)
 (* # Simple functions                                                         *)
@@ -72,28 +73,31 @@ Local Open Scope ring_scope.
 
 Module HBSimple.
 
-HB.structure Definition SimpleFun d (aT : sigmaRingType d) (rT : realType) :=
-  {f of @isMeasurableFun d _ aT rT f & @FiniteImage aT rT f}.
+HB.structure Definition SimpleFun d d'
+    (aT : sigmaRingType d) (rT : sigmaRingType d') :=
+  {f of @isMeasurableFun d d' aT rT f & @FiniteImage aT rT f}.
 
 End HBSimple.
 
-Notation "{ 'sfun' aT >-> T }" := (@HBSimple.SimpleFun.type _ aT T) : form_scope.
+Notation "{ 'sfun' aT >-> T }" :=
+  (@HBSimple.SimpleFun.type _ _ aT T) : form_scope.
 Notation "[ 'sfun' 'of' f ]" := [the {sfun _ >-> _} of f] : form_scope.
 
 Module HBNNSimple.
 Import HBSimple.
 
 HB.structure Definition NonNegSimpleFun
     d (aT : sigmaRingType d) (rT : realType) :=
-  {f of @SimpleFun d _ _ f & @NonNegFun aT rT f}.
+  {f of @SimpleFun d _ _ _ f & @NonNegFun aT rT f}.
 
 End HBNNSimple.
 
-Notation "{ 'nnsfun' aT >-> T }" := (@HBNNSimple.NonNegSimpleFun.type _ aT%type T) : form_scope.
+Notation "{ 'nnsfun' aT >-> T }" :=
+  (@HBNNSimple.NonNegSimpleFun.type _ aT%type T) : form_scope.
 Notation "[ 'nnsfun' 'of' f ]" := [the {nnsfun _ >-> _} of f] : form_scope.
 
 Section sfun_pred.
-Context {d} {aT : sigmaRingType d} {rT : realType}.
+Context {d d'} {aT : sigmaRingType d} {rT : sigmaRingType d'}.
 Definition sfun : {pred _ -> _} := [predI @mfun _ _ aT rT & fimfun].
 Definition sfun_key : pred_key sfun. Proof. exact. Qed.
 Canonical sfun_keyed := KeyedPred sfun_key.
@@ -102,13 +106,13 @@ Lemma sub_sfun_fimfun : {subset sfun <= fimfun}. Proof. by move=> x /andP[]. Qed
 End sfun_pred.
 
 Section sfun.
-Context {d} {aT : measurableType d} {rT : realType}.
+Context {d d'} {aT : measurableType d} {rT : sigmaRingType d'}.
 Notation T := {sfun aT >-> rT}.
-Notation sfun := (@sfun _ aT rT).
+Notation sfun := (@sfun _ _ aT rT).
 Section Sub.
 Context (f : aT -> rT) (fP : f \in sfun).
 Definition sfun_Sub1_subproof :=
-  @isMeasurableFun.Build d _ aT rT f (set_mem (sub_sfun_mfun fP)).
+  @isMeasurableFun.Build d d' aT rT f (set_mem (sub_sfun_mfun fP)).
 #[local] HB.instance Definition _ := sfun_Sub1_subproof.
 Definition sfun_Sub2_subproof :=
   @FiniteImage.Build aT rT f (set_mem (sub_sfun_fimfun fP)).
@@ -150,9 +154,9 @@ Lemma cst_sfunE x : @cst_sfun x =1 cst x. Proof. by []. Qed.
 End sfun.
 
 (* a better way to refactor function stuffs *)
-Lemma fctD (T : Type) (K : pzRingType) (f g : T -> K) : f + g = f \+ g.
+Lemma fctD (T : Type) (N : nmodType) (f g : T -> N) : f + g = f \+ g.
 Proof. by []. Qed.
-Lemma fctN (T : Type) (K : pzRingType) (f : T -> K) : - f = \- f.
+Lemma fctN (T : Type) (N : zmodType) (f : T -> N) : - f = \- f.
 Proof. by []. Qed.
 Lemma fctM (T : Type) (K : pzRingType) (f g : T -> K) : f * g = f \* g.
 Proof. by []. Qed.
@@ -165,13 +169,13 @@ Definition fctWE := (fctD, fctN, fctM, fctZ).
 Section ring.
 Context d (aT : measurableType d) (rT : realType).
 
-Lemma sfun_subring_closed : subring_closed (@sfun d aT rT).
+Lemma sfun_subring_closed : subring_closed (@sfun d _ aT rT).
 Proof.
 by split=> [|f g|f g]; rewrite ?inE/= ?rpred1//;
    move=> /andP[/= mf ff] /andP[/= mg fg]; rewrite !(rpredB, rpredM).
 Qed.
 
-HB.instance Definition _ := GRing.isSubringClosed.Build _ sfun
+HB.instance Definition _ := GRing.isSubringClosed.Build _ (@sfun d _ aT rT)
   sfun_subring_closed.
 HB.instance Definition _ := [SubChoice_isSubComPzRing of {sfun aT >-> rT} by <:].
 
@@ -213,6 +217,50 @@ Definition scale_sfun k f : {sfun aT >-> rT} := k \o* f.
 End ring.
 Arguments indic_sfun {d aT rT} _.
 
+Section sfun_lmodType.
+Context d (aT : measurableType d) (R : realType).
+Import HBSimple.
+
+Lemma mem_sfun_comp_pair (U V W : normedModType R) (f : {sfun aT >-> U})
+    (g : {sfun aT >-> V}) (h : U * V -> W) :
+  (fun x => h (f x, g x)) \in @sfun _ _ aT W.
+Proof.
+rewrite inE; apply/andP; split; rewrite inE/=.
+  move=> _ Y mY; rewrite setTI.
+  rewrite (_ : _ @^-1` Y =
+      \bigcup_(a in range f) (\bigcup_(b in range g)
+        ((f @^-1` [set a] `&` g @^-1` [set b]) `&` [set _ | Y (h (a, b))]))).
+    apply/seteqP; split=> [x/= Yfg|x [a _] [b _] [[/= <- <-]]//].
+    by exists (f x); [exists x|exists (g x); [exists x|]].
+  apply: fin_bigcup_measurable; first exact: fimfunP.
+  move=> a _; apply: fin_bigcup_measurable; first exact: fimfunP.
+  move=> b _; apply: measurableI.
+    by apply: measurableI; exact/(measurable_funPTI _ (measurable1 _)).
+  have [Yhab|Yhab] := pselect (Y (h (a, b))).
+    by rewrite (_ : [set _ | _] = setT);
+        [apply/seteqP; split|exact: measurableT].
+  rewrite (_ : [set _ | _] = set0); last exact: measurable0.
+  by apply/seteqP; split.
+apply: (@sub_finite_set _ _ (h @` (range f `*` range g))).
+  by move=> _ [x _ <-]/=; exists (f x, g x) => //; split; exists x.
+exact/finite_image/finite_setX/fimfunP.
+Qed.
+
+Lemma sfun_submod_closed (V : normedModType R) :
+  submod_closed (@sfun _ _ aT V).
+Proof.
+split=> [|k f g sf sg]; first exact: (valP (cst_sfun (0 : V))).
+exact: (mem_sfun_comp_pair (sfun_Sub sf) (sfun_Sub sg) (fun t => k *: t.1 + t.2)).
+Qed.
+
+HB.instance Definition _ (V : normedModType R) :=
+  GRing.isSubmodClosed.Build _ _ (@sfun _ _ aT V)
+    (sfun_submod_closed V).
+HB.instance Definition _ (V : normedModType R) :=
+  [SubChoice_isSubLmodule of {sfun aT >-> V} by <:].
+
+End sfun_lmodType.
+
 Lemma preimage_nnfun0 T (R : realDomainType) (f : {nnfun T >-> R}) t :
   t < 0 -> f @^-1` [set t] = set0.
 Proof.

theories/measure_theory/measurable_function.v

@@ -5,7 +5,7 @@ From mathcomp Require Import all_ssreflect_compat algebra.
 From mathcomp Require Import unstable.
 From mathcomp Require Import boolp classical_sets functions cardinality reals.
 From mathcomp Require Import ereal topology normedtype.
-From mathcomp Require Import sequences measurable_structure.
+From mathcomp Require Import sequences measurable_structure measurable_topology.
 
 (**md**************************************************************************)
 (* # Measurable Functions                                                     *)
@@ -68,6 +68,9 @@ Proof. by move=> mY; rewrite -[f @^-1` _]setTI; exact: measurable_funP. Qed.
 #[deprecated(since="mathcomp-analysis 1.13.0", note="renamed to `measurable_funPTI`")]
 Notation measurable_sfunP := measurable_funPTI (only parsing).
 
+#[global] Hint Extern 0 (measurable (_ @^-1` [set _])) =>
+  solve [apply: measurable_funPTI; exact: measurable1] : core.
+
 Section mfun_pred.
 Context {d d'} {aT : sigmaRingType d} {rT : sigmaRingType d'}.
 Definition mfun : {pred aT -> rT} := mem [set f | measurable_fun setT f].

theories/measure_theory/measurable_topology.v

@@ -0,0 +1,52 @@
+(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C.              *)
+From HB Require Import structures.
+From mathcomp Require Import all_ssreflect_compat algebra finmap.
+#[warning="-warn-library-file-internal-analysis"]
+From mathcomp Require Import unstable.
+From mathcomp Require Import boolp classical_sets functions cardinality reals topology.
+From mathcomp Require Import measurable_structure normedtype.
+
+Unset SsrOldRewriteGoalsOrder.  (* remove the line when requiring MathComp >= 2.6 *)
+Set Implicit Arguments.
+Unset Strict Implicit.
+Unset Printing Implicit Defensive.
+
+Import Order.TTheory GRing.Theory Num.Def Num.Theory.
+Import numFieldTopology.Exports.
+
+Local Open Scope classical_set_scope.
+
+Definition borelType (T : topologicalType) := g_sigma_algebraType (@open T).
+
+Section borel_normedModType.
+Local Open Scope ring_scope.
+Context {R : realType} {V : normedModType R}.
+
+Lemma singleton_bigcap (x : V) :
+  [set x] = \bigcap_(k : nat) ball x (k.+1%:R)^-1.
+Proof.
+apply/seteqP; split => [_ -> k _|y xy].
+  by rewrite -ball_normE/= subrr normr0 invr_gt0 ltr0n.
+apply/eqP; rewrite eq_sym -subr_eq0 -normr_eq0 eq_le normr_ge0 andbT.
+apply/ler_addgt0Pl => e e0; rewrite addr0.
+have := xy (truncn e^-1) I; rewrite -ball_normE/= => /ltW/le_trans; apply.
+by rewrite invf_ple ?posrE ?ltr0n ?invr_gt0//; apply/ltW/truncnS_gt.
+Qed.
+
+Lemma measurable1 (x : borelType V) : measurable [set x].
+Proof.
+rewrite singleton_bigcap; apply: bigcap_measurable => // k _.
+by apply: sub_sigma_algebra; exact: ball_open.
+Qed.
+#[local] Hint Resolve measurable1 : core.
+
+End borel_normedModType.
+
+#[global]
+Hint Extern 0 (measurable [set _]) => solve [apply: measurable1] : core.
+
+#[non_forgetful_inheritance]
+HB.instance Definition _ (R : realType) (V : normedModType R) :=
+  Measurable.copy V (borelType V).
+
+