Module mathcomp.analysis.kernel
From HB Require Import structures.From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality fsbigop.
Require Import reals ereal signed topology normedtype sequences esum measure.
Require Import numfun lebesgue_measure lebesgue_integral.
# Kernels
This file provides a formation of kernels, s-finite kernels, finite
kernels, subprobability kernels, and probability kernels. The main
formalized result is the fact that s-finite kernels are stable by
composition.
Reference:
- R. Affeldt, C. Cohen, A. Saito. Semantics of probabilistic programs
using s-finite kernels in Coq. CPP 2023
```
R.-ker X ~> Y == kernel from X to Y where X and Y are of type
measurableType
The HB class is Kernel.
measure_fam_uub k == the kernel k is uniformly upper-bounded
R.-sfker X ~> Y == s-finite kernel
The HB class is SFiniteKernel.
R.-fker X ~> Y == finite kernel
The HB class is FiniteKernel.
R.-spker X ~> Y == subprobability kernel
The HB class is SubProbabilityKernel.
R.-pker X ~> Y == probability kernel
The HB class is ProbabilityKernel.
kseries == countable sum of kernels
It is declared as an instance of the structure
Kernel. It is also an instance of the structure
SFiniteKernel if the sum is over s-finite kernels.
kzero == kernel defined using the mzero measure
kdirac mf == kernel defined by a measurable function
kprobability m == kernel defined by a probability measure
kadd k1 k2 == lifting of the addition of measures to kernels
l \; k == composition of kernels
```
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.
Local Open Scope ring_scope.
Local Open Scope ereal_scope.
Reserved Notation "R .-ker X ~> Y"
(at level 42, format "R .-ker X ~> Y").
Reserved Notation "R .-sfker X ~> Y"
(at level 42, format "R .-sfker X ~> Y").
Reserved Notation "R .-fker X ~> Y"
(at level 42, format "R .-fker X ~> Y").
Reserved Notation "R .-spker X ~> Y"
(at level 42, format "R .-spker X ~> Y").
Reserved Notation "R .-pker X ~> Y"
(at level 42, format "R .-pker X ~> Y").
HB.mixin Record isKernel d d' (X : measurableType d) (Y : measurableType d')
(R : realType) (k : X -> {measure set Y -> \bar R}) := {
measurable_kernel :
forall U, measurable U -> measurable_fun [set: X] (k ^~ U) }.
#[short(type=kernel)]
HB.structure Definition Kernel d d'
(X : measurableType d) (Y : measurableType d') (R : realType) :=
{ k & isKernel _ _ X Y R k }.
Notation "R .-ker X ~> Y" := (kernel X%type Y R).
Arguments measurable_kernel {_ _ _ _ _} _.
Lemma kernel_measurable_eq_cst d d' (T : measurableType d)
(T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k :
measurable [set t | f t [set: T'] == k].
Proof.
rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set k]); last first.
by apply/seteqP; split => t/= /eqP.
have /(_ measurableT [set k]) := measurable_kernel f setT measurableT.
by rewrite setTI; exact.
Qed.
by apply/seteqP; split => t/= /eqP.
have /(_ measurableT [set k]) := measurable_kernel f setT measurableT.
by rewrite setTI; exact.
Qed.
Lemma kernel_measurable_neq_cst d d' (T : measurableType d)
(T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k :
measurable [set t | f t [set: T'] != k].
Proof.
rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set~ k]); last first.
by apply/seteqP; split => t /eqP.
have /(_ measurableT [set~ k]) := measurable_kernel f setT measurableT.
by rewrite setTI; apply => //; exact: measurableC.
Qed.
by apply/seteqP; split => t /eqP.
have /(_ measurableT [set~ k]) := measurable_kernel f setT measurableT.
by rewrite setTI; apply => //; exact: measurableC.
Qed.
Lemma kernel_measurable_fun_eq_cst d d' (T : measurableType d)
(T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k :
measurable_fun [set: T] (fun t => f t [set: T'] == k).
Proof.
move=> _ /= B mB; rewrite setTI.
have [/eqP->|/eqP->|/eqP->|/eqP->] := set_bool B.
- exact: kernel_measurable_eq_cst.
- rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first.
by apply/seteqP; split => [t /negbT//|t /negbTE].
exact: kernel_measurable_neq_cst.
- by rewrite preimage_set0.
- by rewrite preimage_setT.
Qed.
have [/eqP->|/eqP->|/eqP->|/eqP->] := set_bool B.
- exact: kernel_measurable_eq_cst.
- rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first.
by apply/seteqP; split => [t /negbT//|t /negbTE].
exact: kernel_measurable_neq_cst.
- by rewrite preimage_set0.
- by rewrite preimage_setT.
Qed.
Lemma eq_kernel d d' (T : measurableType d) (T' : measurableType d')
(R : realType) (k1 k2 : R.-ker T ~> T') :
(forall x U, k1 x U = k2 x U) -> k1 = k2.
Proof.
move: k1 k2 => [m1 [[?]]] [m2 [[?]]] /= k12.
have ? : m1 = m2.
by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12.
by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance.
Qed.
have ? : m1 = m2.
by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12.
by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance.
Qed.
Section kseries.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable k : (R.-ker X ~> Y)^nat.
Definition kseries : X -> {measure set Y -> \bar R} :=
fun x => [the measure _ _ of mseries (k ^~ x) 0].
Lemma measurable_fun_kseries (U : set Y) :
measurable U -> measurable_fun [set: X] (kseries ^~ U).
Proof.
HB.instance Definition _ :=
isKernel.Build _ _ _ _ _ kseries measurable_fun_kseries.
End kseries.
Lemma integral_kseries d d' (X : measurableType d) (Y : measurableType d')
(R : realType) (k : (R.-ker X ~> Y)^nat) (f : Y -> \bar R) x :
(forall y, 0 <= f y) ->
measurable_fun [set: Y] f ->
\int[kseries k x]_y (f y) = \sum_(i <oo) \int[k i x]_y (f y).
Proof.
Section measure_fam_uub.
Context d d' (X : measurableType d) (Y : measurableType d') (R : numFieldType).
Variable k : X -> {measure set Y -> \bar R}.
Definition measure_fam_uub := exists r, forall x, k x [set: Y] < r%:E.
Lemma measure_fam_uubP : measure_fam_uub <->
exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E.
Proof.
End measure_fam_uub.
HB.mixin Record Kernel_isSFinite_subdef d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) := {
sfinite_kernel_subdef : exists2 s : (R.-ker X ~> Y)^nat,
forall n, measure_fam_uub (s n) &
forall x U, measurable U -> k x U = kseries s x U }.
HB.structure Definition SFiniteKernel d d'
(X : measurableType d) (Y : measurableType d') (R : realType) :=
{ k of @Kernel _ _ _ _ R k &
Kernel_isSFinite_subdef _ _ X Y R k }.
Notation "R .-sfker X ~> Y" := (SFiniteKernel.type X%type Y R).
Arguments sfinite_kernel_subdef {_ _ _ _ _} _.
Lemma eq_sfkernel d d' (T : measurableType d) (T' : measurableType d')
(R : realType) (k1 k2 : R.-sfker T ~> T') :
(forall x U, k1 x U = k2 x U) -> k1 = k2.
Proof.
move: k1 k2 => [m1 [[?] [?]]] [m2 [[?] [?]]] /= k12.
have ? : m1 = m2.
by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12.
by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance.
Qed.
have ? : m1 = m2.
by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12.
by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance.
Qed.
HB.mixin Record SFiniteKernel_isFinite d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) := {
measure_uub : measure_fam_uub k }.
#[short(type=finite_kernel)]
HB.structure Definition FiniteKernel d d'
(X : measurableType d) (Y : measurableType d') (R : realType) :=
{ k of @SFiniteKernel _ _ _ _ _ k &
SFiniteKernel_isFinite _ _ X Y R k }.
Notation "R .-fker X ~> Y" := (finite_kernel X%type Y R).
Arguments measure_uub {_ _ _ _ _} _.
HB.factory Record Kernel_isFinite d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) of isKernel _ _ _ _ _ k := {
measure_uub : measure_fam_uub k }.
Section kzero.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Definition kzero : X -> {measure set Y -> \bar R} :=
fun _ : X => [the measure _ _ of mzero].
Let measurable_fun_kzero U : measurable U ->
measurable_fun [set: X] (kzero ^~ U).
Proof.
HB.instance Definition _ :=
@isKernel.Build _ _ X Y R kzero measurable_fun_kzero.
Lemma kzero_uub : measure_fam_uub kzero.
Proof.
End kzero.
HB.builders Context d d' (X : measurableType d) (Y : measurableType d')
(R : realType) k of Kernel_isFinite d d' X Y R k.
Let sfinite_finite :
exists2 k_ : (R.-ker _ ~> _)^nat, forall n, measure_fam_uub (k_ n) &
forall x U, measurable U -> k x U = mseries (k_ ^~ x) 0 U.
Proof.
exists (fun n => if n is O then [the _.-ker _ ~> _ of k] else
[the _.-ker _ ~> _ of @kzero _ _ X Y R]).
by case => [|_]; [exact: measure_uub|exact: kzero_uub].
move=> t U mU/=; rewrite /mseries.
rewrite (nneseries_split _ 1%N)// big_mkord big_ord_recl/= big_ord0 adde0.
rewrite ereal_series (@eq_eseriesr _ _ (fun=> 0%E)); last by case.
by rewrite eseries0// adde0.
Qed.
[the _.-ker _ ~> _ of @kzero _ _ X Y R]).
by case => [|_]; [exact: measure_uub|exact: kzero_uub].
move=> t U mU/=; rewrite /mseries.
rewrite (nneseries_split _ 1%N)// big_mkord big_ord_recl/= big_ord0 adde0.
rewrite ereal_series (@eq_eseriesr _ _ (fun=> 0%E)); last by case.
by rewrite eseries0// adde0.
Qed.
HB.instance Definition _ :=
@Kernel_isSFinite_subdef.Build d d' X Y R k sfinite_finite.
HB.instance Definition _ :=
@SFiniteKernel_isFinite.Build d d' X Y R k measure_uub.
HB.end.
Section sfinite.
Context d d' (X : measurableType d) (Y : measurableType d').
Variables (R : realType) (k : R.-sfker X ~> Y).
Let s : (X -> {measure set Y -> \bar R})^nat :=
let: exist2 x _ _ := cid2 (sfinite_kernel_subdef k) in x.
Let ms n : @isKernel d d' X Y R (s n).
Proof.
HB.instance Definition _ n := ms n.
Let s_uub n : measure_fam_uub (s n).
HB.instance Definition _ n := @Kernel_isFinite.Build d d' X Y R (s n) (s_uub n).
Lemma sfinite_kernel : exists s : (R.-fker X ~> Y)^nat,
forall x U, measurable U -> k x U = kseries s x U.
Proof.
End sfinite.
Lemma sfinite_kernel_measure d d' (Z : measurableType d) (X : measurableType d')
(R : realType) (k : R.-sfker Z ~> X) (z : Z) :
sfinite_measure (k z).
Proof.
have [s ks] := sfinite_kernel k.
exists (s ^~ z).
move=> n; have [r snr] := measure_uub (s n).
by apply: lty_fin_num_fun; rewrite (lt_le_trans (snr _))// leey.
by move=> U mU; rewrite ks.
Qed.
exists (s ^~ z).
move=> n; have [r snr] := measure_uub (s n).
by apply: lty_fin_num_fun; rewrite (lt_le_trans (snr _))// leey.
by move=> U mU; rewrite ks.
Qed.
HB.instance Definition _
d d' (X : measurableType d) (Y : measurableType d') (R : realType) :=
@Kernel_isFinite.Build _ _ _ _ R (@kzero _ _ X Y R)
(@kzero_uub _ _ X Y R).
HB.factory Record Kernel_isSFinite d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) of isKernel _ _ _ _ _ k := {
sfinite : exists s : (R.-fker X ~> Y)^nat,
forall x U, measurable U -> k x U = kseries s x U }.
HB.builders Context d d' (X : measurableType d) (Y : measurableType d')
(R : realType) k of Kernel_isSFinite d d' X Y R k.
Lemma sfinite_subdef : Kernel_isSFinite_subdef d d' X Y R k.
Proof.
HB.instance Definition _ := sfinite_subdef.
HB.end.
Section sfkseries.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variables k : (R.-sfker X ~> Y)^nat.
Let sfinite_kseries : exists2 k_ : (R.-ker _ ~> _)^nat,
forall n, measure_fam_uub (k_ n) &
forall x U, measurable U -> kseries k x U = mseries (k_ ^~ x) 0 U.
Proof.
have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card.
by rewrite card_eq_sym; exact: card_nat2.
pose p n : (R.-fker X ~> Y)^nat := sval (cid (sfinite_kernel (k n))).
exists (fun i => p (f i).1 (f i).2).
move=> n; have [r Hr] := measure_uub (p (f n).1 (f n).2).
by exists r => x /=; exact: Hr.
move=> x U mU; rewrite /kseries /mseries/= /mseries/=.
have kE i : k i x U = \sum_(j <oo) p i j x U.
by rewrite /p /sval/=; case: cid => k_ ->.
transitivity (\esum_(l in [set: nat] `*` [set: nat]) p l.1 l.2 x U).
rewrite (_ : _ `*` _ = setT `*`` (fun=> setT)); last by apply/seteqP; split.
rewrite -(@esum_esum _ _ _ _ _ (fun i j => p i j x U))//.
rewrite nneseries_esum// -fun_true; apply: eq_esum => i _.
by rewrite kE// nneseries_esum.
rewrite (reindex_esum [set: nat] _ f)//; last first.
have := @bijTT _ _ f.
by rewrite -setTT_bijective/= -[in X in set_bij _ X _ -> _](@setXTT nat nat).
by rewrite nneseries_esum// fun_true; exact: eq_esum.
Qed.
by rewrite card_eq_sym; exact: card_nat2.
pose p n : (R.-fker X ~> Y)^nat := sval (cid (sfinite_kernel (k n))).
exists (fun i => p (f i).1 (f i).2).
move=> n; have [r Hr] := measure_uub (p (f n).1 (f n).2).
by exists r => x /=; exact: Hr.
move=> x U mU; rewrite /kseries /mseries/= /mseries/=.
have kE i : k i x U = \sum_(j <oo) p i j x U.
by rewrite /p /sval/=; case: cid => k_ ->.
transitivity (\esum_(l in [set: nat] `*` [set: nat]) p l.1 l.2 x U).
rewrite (_ : _ `*` _ = setT `*`` (fun=> setT)); last by apply/seteqP; split.
rewrite -(@esum_esum _ _ _ _ _ (fun i j => p i j x U))//.
rewrite nneseries_esum// -fun_true; apply: eq_esum => i _.
by rewrite kE// nneseries_esum.
rewrite (reindex_esum [set: nat] _ f)//; last first.
have := @bijTT _ _ f.
by rewrite -setTT_bijective/= -[in X in set_bij _ X _ -> _](@setXTT nat nat).
by rewrite nneseries_esum// fun_true; exact: eq_esum.
Qed.
HB.instance Definition _ :=
Kernel_isSFinite_subdef.Build _ _ _ _ R (kseries k) sfinite_kseries.
End sfkseries.
HB.mixin Record FiniteKernel_isSubProbability d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) := {
sprob_kernel : ereal_sup [set k x [set: Y] | x in [set: X]] <= 1 }.
#[short(type=sprobability_kernel)]
HB.structure Definition SubProbabilityKernel
d d' (X : measurableType d) (Y : measurableType d') (R : realType) :=
{ k of @FiniteKernel _ _ _ _ _ k &
FiniteKernel_isSubProbability _ _ X Y R k }.
Notation "R .-spker X ~> Y" := (sprobability_kernel X%type Y R).
HB.factory Record Kernel_isSubProbability d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := {
sprob_kernel : ereal_sup [set k x [set: Y] | x in [set: X]] <= 1 }.
HB.builders Context d d' (X : measurableType d) (Y : measurableType d')
(R : realType) k of Kernel_isSubProbability d d' X Y R k.
Let finite : @Kernel_isFinite d d' X Y R k.
Proof.
split; exists 2%R => /= ?; rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n//.
by rewrite (le_trans _ sprob_kernel)//; exact: ereal_sup_ubound.
Qed.
by rewrite (le_trans _ sprob_kernel)//; exact: ereal_sup_ubound.
Qed.
HB.instance Definition _ := finite.
HB.instance Definition _ :=
@FiniteKernel_isSubProbability.Build _ _ _ _ _ k sprob_kernel.
HB.end.
HB.mixin Record SubProbability_isProbability d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) := {
prob_kernel : forall x, k x [set: Y] = 1 }.
#[short(type=probability_kernel)]
HB.structure Definition ProbabilityKernel d d'
(X : measurableType d) (Y : measurableType d') (R : realType) :=
{ k of @SubProbabilityKernel _ _ _ _ _ k &
SubProbability_isProbability _ _ X Y R k }.
Notation "R .-pker X ~> Y" := (probability_kernel X%type Y R).
HB.factory Record Kernel_isProbability d d'
(X : measurableType d) (Y : measurableType d') (R : realType)
(k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := {
prob_kernel : forall x, k x [set: Y] = 1 }.
HB.builders Context d d' (X : measurableType d) (Y : measurableType d')
(R : realType) k of Kernel_isProbability d d' X Y R k.
Let sprob_kernel : @Kernel_isSubProbability d d' X Y R k.
Proof.
HB.instance Definition _ := sprob_kernel.
HB.instance Definition _ :=
@SubProbability_isProbability.Build _ _ _ _ _ k prob_kernel.
HB.end.
Lemma finite_kernel_measure d d' (X : measurableType d)
(Y : measurableType d') (R : realType) (k : R.-fker X ~> Y) (x : X) :
fin_num_fun (k x).
Proof.
Section measurable_prod_subset_kernel.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Implicit Types A : set (X * Y).
Section xsection_kernel.
Variable (k : R.-ker X ~> Y) (D : set Y) (mD : measurable D).
Let kD x := mrestr (k x) mD.
HB.instance Definition _ x := Measure.on (kD x).
Let phi A := fun x => kD x (xsection A x).
Let XY := [set A | measurable A /\ measurable_fun [set: X] (phi A)].
Let phiM (A : set X) B : phi (A `*` B) = (fun x => kD x B * (\1_A x)%:E).
Proof.
Lemma measurable_prod_subset_xsection_kernel :
(forall x, exists M, forall X, measurable X -> kD x X < M%:E) ->
measurable `<=` XY.
Proof.
move=> kD_ub; rewrite measurable_prod_measurableType.
set C := [set A `*` B | A in measurable & B in measurable].
have CI : setI_closed C.
move=> _ _ [X1 mX1 [X2 mX2 <-]] [Y1 mY1 [Y2 mY2 <-]].
exists (X1 `&` Y1); first exact: measurableI.
by exists (X2 `&` Y2); [exact: measurableI|rewrite setXI].
have CT : C setT by exists setT => //; exists setT => //; rewrite setXTT.
have CXY : C `<=` XY.
move=> _ [A mA [B mB <-]]; split; first exact: measurableX.
rewrite phiM.
apply: emeasurable_funM => //; first exact/measurable_kernel/measurableI.
by apply/EFin_measurable_fun; rewrite (_ : \1_ _ = mindic R mA).
suff lsystemB : lambda_system setT XY by exact: lambda_system_subset.
split => //; [exact: CXY| |exact: xsection_ndseq_closed].
move=> A B BA [mA mphiA] [mB mphiB]; split; first exact: measurableD.
suff : phi (A `\` B) = (fun x => phi A x - phi B x).
by move=> ->; exact: emeasurable_funB.
rewrite funeqE => x; rewrite /phi/= xsectionD// measureD.
- by rewrite setIidr//; exact: le_xsection.
- exact: measurable_xsection.
- exact: measurable_xsection.
- have [M kM] := kD_ub x.
rewrite (lt_le_trans (kM (xsection A x) _)) ?leey//.
exact: measurable_xsection.
Qed.
set C := [set A `*` B | A in measurable & B in measurable].
have CI : setI_closed C.
move=> _ _ [X1 mX1 [X2 mX2 <-]] [Y1 mY1 [Y2 mY2 <-]].
exists (X1 `&` Y1); first exact: measurableI.
by exists (X2 `&` Y2); [exact: measurableI|rewrite setXI].
have CT : C setT by exists setT => //; exists setT => //; rewrite setXTT.
have CXY : C `<=` XY.
move=> _ [A mA [B mB <-]]; split; first exact: measurableX.
rewrite phiM.
apply: emeasurable_funM => //; first exact/measurable_kernel/measurableI.
by apply/EFin_measurable_fun; rewrite (_ : \1_ _ = mindic R mA).
suff lsystemB : lambda_system setT XY by exact: lambda_system_subset.
split => //; [exact: CXY| |exact: xsection_ndseq_closed].
move=> A B BA [mA mphiA] [mB mphiB]; split; first exact: measurableD.
suff : phi (A `\` B) = (fun x => phi A x - phi B x).
by move=> ->; exact: emeasurable_funB.
rewrite funeqE => x; rewrite /phi/= xsectionD// measureD.
- by rewrite setIidr//; exact: le_xsection.
- exact: measurable_xsection.
- exact: measurable_xsection.
- have [M kM] := kD_ub x.
rewrite (lt_le_trans (kM (xsection A x) _)) ?leey//.
exact: measurable_xsection.
Qed.
End xsection_kernel.
End measurable_prod_subset_kernel.
Section measurable_fun_xsection_finite_kernel.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable k : R.-fker X ~> Y.
Implicit Types A : set (X * Y).
Let phi A := fun x => k x (xsection A x).
Let XY := [set A | measurable A /\ measurable_fun [set: X] (phi A)].
Lemma measurable_fun_xsection_finite_kernel A :
A \in measurable -> measurable_fun [set: X] (phi A).
Proof.
move: A; suff : measurable `<=` XY by move=> + A; rewrite inE => /[apply] -[].
move=> /= A mA; rewrite /XY/=; split => //; rewrite (_ : phi _ =
(fun x => mrestr (k x) measurableT (xsection A x))); last first.
by apply/funext => x//=; rewrite /mrestr setIT.
apply measurable_prod_subset_xsection_kernel => // x.
have [r hr] := measure_uub k; exists r => B mB.
by rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT le_measure// inE.
Qed.
move=> /= A mA; rewrite /XY/=; split => //; rewrite (_ : phi _ =
(fun x => mrestr (k x) measurableT (xsection A x))); last first.
by apply/funext => x//=; rewrite /mrestr setIT.
apply measurable_prod_subset_xsection_kernel => // x.
have [r hr] := measure_uub k; exists r => B mB.
by rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT le_measure// inE.
Qed.
End measurable_fun_xsection_finite_kernel.
Section measurable_fun_integral_finite_sfinite.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable k : X * Y -> \bar R.
Lemma measurable_fun_xsection_integral
(l : X -> {measure set Y -> \bar R})
(k_ : {nnsfun (X * Y) >-> R}^nat)
(ndk_ : nondecreasing_seq (k_ : (X * Y -> R)^nat))
(k_k : forall z, (k_ n z)%:E @[n --> \oo] --> k z) :
(forall n r,
measurable_fun [set: X] (fun x => l x (xsection (k_ n @^-1` [set r]) x))) ->
measurable_fun [set: X] (fun x => \int[l x]_y k (x, y)).
Proof.
move=> h.
rewrite (_ : (fun x => _) =
(fun x => limn_esup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first.
apply/funext => x.
transitivity (lim (\int[l x]_y (k_ n (x, y))%:E @[n --> \oo])); last first.
rewrite is_cvg_limn_esupE//.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: ge0_le_integral => //.
- by move=> y _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> y _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> y _; rewrite lee_fin; exact/lefP/ndk_.
rewrite -monotone_convergence//.
- by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k.
- by move=> n; exact/EFin_measurable_fun/measurableT_comp.
- by move=> n y _; rewrite lee_fin.
- by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_.
apply: measurable_fun_limn_esup => n.
rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y
(\sum_(r \in range (k_ n))
r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first.
by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE.
rewrite [X in measurable_fun _ X](_ : _ = (fun x => \sum_(r \in range (k_ n))
(\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first.
apply/funext => x; rewrite -ge0_integral_fsum//.
- by apply: eq_integral => y _; rewrite -fsumEFin.
- move=> r.
apply/EFin_measurable_fun/measurableT_comp => [//|].
exact/measurableT_comp.
- by move=> m y _; rewrite nnfun_muleindic_ge0.
apply: emeasurable_fun_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.
have [r0|r0] := leP 0%R r.
rewrite ge0_integralZl//; last by move=> *; rewrite lee_fin.
exact/EFin_measurable_fun/measurableT_comp.
rewrite integral0_eq; last first.
by move=> y _; rewrite preimage_nnfun0// indic0 mule0.
by rewrite integral0_eq ?mule0// => y _; rewrite preimage_nnfun0// indic0.
apply/measurable_funeM.
rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))).
exact/h.
apply/funext => x; rewrite integral_indic//; last first.
rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) x).
exact: measurable_xsection.
by rewrite /xsection; apply/seteqP; split=> y/= /[!inE].
congr (l x _); apply/funext => y1/=; rewrite /xsection/= inE.
by apply/propext; tauto.
Qed.
rewrite (_ : (fun x => _) =
(fun x => limn_esup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first.
apply/funext => x.
transitivity (lim (\int[l x]_y (k_ n (x, y))%:E @[n --> \oo])); last first.
rewrite is_cvg_limn_esupE//.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: ge0_le_integral => //.
- by move=> y _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> y _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> y _; rewrite lee_fin; exact/lefP/ndk_.
rewrite -monotone_convergence//.
- by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k.
- by move=> n; exact/EFin_measurable_fun/measurableT_comp.
- by move=> n y _; rewrite lee_fin.
- by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_.
apply: measurable_fun_limn_esup => n.
rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y
(\sum_(r \in range (k_ n))
r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first.
by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE.
rewrite [X in measurable_fun _ X](_ : _ = (fun x => \sum_(r \in range (k_ n))
(\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first.
apply/funext => x; rewrite -ge0_integral_fsum//.
- by apply: eq_integral => y _; rewrite -fsumEFin.
- move=> r.
apply/EFin_measurable_fun/measurableT_comp => [//|].
exact/measurableT_comp.
- by move=> m y _; rewrite nnfun_muleindic_ge0.
apply: emeasurable_fun_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.
have [r0|r0] := leP 0%R r.
rewrite ge0_integralZl//; last by move=> *; rewrite lee_fin.
exact/EFin_measurable_fun/measurableT_comp.
rewrite integral0_eq; last first.
by move=> y _; rewrite preimage_nnfun0// indic0 mule0.
by rewrite integral0_eq ?mule0// => y _; rewrite preimage_nnfun0// indic0.
apply/measurable_funeM.
rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))).
exact/h.
apply/funext => x; rewrite integral_indic//; last first.
rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) x).
exact: measurable_xsection.
by rewrite /xsection; apply/seteqP; split=> y/= /[!inE].
congr (l x _); apply/funext => y1/=; rewrite /xsection/= inE.
by apply/propext; tauto.
Qed.
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].
Qed.
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].
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].
Qed.
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].
Qed.
End measurable_fun_integral_finite_sfinite.
Arguments measurable_fun_xsection_integral {_ _ _ _ _} k l.
Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} k l.
Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} k l.
Section kdirac.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable f : X -> Y.
Definition kdirac (mf : measurable_fun [set: X] f)
: X -> {measure set Y -> \bar R} :=
fun x => [the measure _ _ of dirac (f x)].
Hypothesis mf : measurable_fun [set: X] f.
Let measurable_fun_kdirac U : measurable U ->
measurable_fun [set: X] (kdirac mf ^~ U).
Proof.
move=> mU; apply/EFin_measurable_fun.
by rewrite (_ : (fun x => _) = mindic R mU \o f)//; exact/measurableT_comp.
Qed.
by rewrite (_ : (fun x => _) = mindic R mU \o f)//; exact/measurableT_comp.
Qed.
HB.instance Definition _ := isKernel.Build _ _ _ _ _ (kdirac mf)
measurable_fun_kdirac.
Let kdirac_prob x : kdirac mf x [set: Y] = 1.
HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _
(kdirac mf) kdirac_prob.
End kdirac.
Arguments kdirac {d d' X Y R f}.
Section kprobability.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable P : X -> pprobability Y R.
Definition kprobability (mP : measurable_fun [set: X] P)
: X -> {measure set Y -> \bar R} := P.
Hypothesis mP : measurable_fun [set: X] P.
Let measurable_fun_kprobability U : measurable U ->
measurable_fun [set: X] (kprobability mP ^~ U).
Proof.
move=> mU.
apply: (measurability (ErealGenInftyO.measurableE R)) => _ /= -[_ [r ->] <-].
rewrite setTI preimage_itv_infty_o -/(P @^-1` mset U r).
have [r0|r0] := leP 0%R r; last by rewrite lt0_mset// preimage_set0.
have [r1|r1] := leP r 1%R; last by rewrite gt1_mset// preimage_setT.
move: mP => /(_ measurableT (mset U r)); rewrite setTI; apply.
by apply: sub_sigma_algebra; exists r => /=; [rewrite in_itv/= r0|exists U].
Qed.
apply: (measurability (ErealGenInftyO.measurableE R)) => _ /= -[_ [r ->] <-].
rewrite setTI preimage_itv_infty_o -/(P @^-1` mset U r).
have [r0|r0] := leP 0%R r; last by rewrite lt0_mset// preimage_set0.
have [r1|r1] := leP r 1%R; last by rewrite gt1_mset// preimage_setT.
move: mP => /(_ measurableT (mset U r)); rewrite setTI; apply.
by apply: sub_sigma_algebra; exists r => /=; [rewrite in_itv/= r0|exists U].
Qed.
HB.instance Definition _ :=
@isKernel.Build _ _ X Y R (kprobability mP) measurable_fun_kprobability.
Let kprobability_prob x : kprobability mP x [set: Y] = 1.
Proof.
HB.instance Definition _ :=
@Kernel_isProbability.Build _ _ X Y R (kprobability mP) kprobability_prob.
End kprobability.
Section kadd.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variables k1 k2 : R.-ker X ~> Y.
Definition kadd : X -> {measure set Y -> \bar R} :=
fun x => [the measure _ _ of measure_add (k1 x) (k2 x)].
Let measurable_fun_kadd U : measurable U ->
measurable_fun [set: X] (kadd ^~ U).
Proof.
move=> mU; rewrite /kadd.
rewrite (_ : (fun _ => _) = (fun x => k1 x U + k2 x U)); last first.
by apply/funext => x; rewrite -measure_addE.
by apply: emeasurable_funD; exact/measurable_kernel.
Qed.
rewrite (_ : (fun _ => _) = (fun x => k1 x U + k2 x U)); last first.
by apply/funext => x; rewrite -measure_addE.
by apply: emeasurable_funD; exact/measurable_kernel.
Qed.
HB.instance Definition _ :=
@isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd.
End kadd.
Section sfkadd.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variables k1 k2 : R.-sfker X ~> Y.
Let sfinite_kadd : exists2 k_ : (R.-ker _ ~> _)^nat,
forall n, measure_fam_uub (k_ n) &
forall x U, measurable U ->
kadd k1 k2 x U = mseries (k_ ^~ x) 0 U.
Proof.
have [f1 hk1] := sfinite_kernel k1; have [f2 hk2] := sfinite_kernel k2.
exists (fun n => [the _.-ker _ ~> _ of kadd (f1 n) (f2 n)]).
move=> n.
have [r1 f1r1] := measure_uub (f1 n).
have [r2 f2r2] := measure_uub (f2 n).
exists (r1 + r2)%R => x/=.
by rewrite /msum !big_ord_recr/= big_ord0 add0e EFinD lteD.
move=> x U mU.
rewrite /kadd/= -/(measure_add (k1 x) (k2 x)) measure_addE hk1//= hk2//=.
rewrite /mseries -nneseriesD//; apply: eq_eseriesr => n _ /=.
by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE.
Qed.
exists (fun n => [the _.-ker _ ~> _ of kadd (f1 n) (f2 n)]).
move=> n.
have [r1 f1r1] := measure_uub (f1 n).
have [r2 f2r2] := measure_uub (f2 n).
exists (r1 + r2)%R => x/=.
by rewrite /msum !big_ord_recr/= big_ord0 add0e EFinD lteD.
move=> x U mU.
rewrite /kadd/= -/(measure_add (k1 x) (k2 x)) measure_addE hk1//= hk2//=.
rewrite /mseries -nneseriesD//; apply: eq_eseriesr => n _ /=.
by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE.
Qed.
HB.instance Definition _ t :=
Kernel_isSFinite_subdef.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd.
End sfkadd.
Section fkadd.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variables k1 k2 : R.-fker X ~> Y.
Let kadd_finite_uub : measure_fam_uub (kadd k1 k2).
Proof.
have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2.
exists (f1 + f2)%R => x; rewrite /kadd /=.
rewrite -/(measure_add (k1 x) (k2 x)).
by rewrite measure_addE EFinD; exact: lteD.
Qed.
exists (f1 + f2)%R => x; rewrite /kadd /=.
rewrite -/(measure_add (k1 x) (k2 x)).
by rewrite measure_addE EFinD; exact: lteD.
Qed.
HB.instance Definition _ t :=
Kernel_isFinite.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub.
End fkadd.
Section knormalize.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variable f : R.-ker X ~> Y.
Definition knormalize (P : probability Y R) : X -> {measure set Y -> \bar R} :=
fun x => mnormalize (f x) P.
Let measurable_knormalize (P : probability Y R) U :
measurable U -> measurable_fun [set: X] (knormalize P ^~ U).
Proof.
move=> mU; rewrite /knormalize/= /mnormalize /=.
rewrite (_ : (fun _ => _) = (fun x =>
if f x setT == 0 then P U else if f x setT == +oo then P U
else f x U * (fine (f x setT))^-1%:E)); last first.
apply/funext => x; case: ifPn => [/orP[->//|->]|]; first by case: ifPn.
by rewrite negb_or=> /andP[/negbTE -> /negbTE ->].
apply: measurable_fun_if => //; [exact: kernel_measurable_fun_eq_cst|].
apply: measurable_fun_if => //.
- rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]).
exact: kernel_measurable_neq_cst.
by apply/seteqP; split => [x /negbT//|x /negbTE].
- apply: (@measurable_funS _ _ _ _ setT) => //.
exact: kernel_measurable_fun_eq_cst.
- apply: emeasurable_funM.
exact: measurable_funS (measurable_kernel f U mU).
apply/EFin_measurable_fun.
apply: (@measurable_comp _ _ _ _ _ _ [set r : R | r != 0%R]) => //.
+ exact: open_measurable.
+ move=> /= r [t] [] [_ ft0] ftoo ftr; apply/eqP => r0.
move: (ftr); rewrite r0 => /eqP; rewrite fine_eq0 ?ft0//.
by rewrite ge0_fin_numE// lt_neqAle leey ftoo.
+ apply: open_continuous_measurable_fun => //; apply/in_setP => x /= x0.
exact: inv_continuous.
+ apply: measurableT_comp => //=.
by have := measurable_kernel f _ measurableT; exact: measurable_funS.
Qed.
rewrite (_ : (fun _ => _) = (fun x =>
if f x setT == 0 then P U else if f x setT == +oo then P U
else f x U * (fine (f x setT))^-1%:E)); last first.
apply/funext => x; case: ifPn => [/orP[->//|->]|]; first by case: ifPn.
by rewrite negb_or=> /andP[/negbTE -> /negbTE ->].
apply: measurable_fun_if => //; [exact: kernel_measurable_fun_eq_cst|].
apply: measurable_fun_if => //.
- rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]).
exact: kernel_measurable_neq_cst.
by apply/seteqP; split => [x /negbT//|x /negbTE].
- apply: (@measurable_funS _ _ _ _ setT) => //.
exact: kernel_measurable_fun_eq_cst.
- apply: emeasurable_funM.
exact: measurable_funS (measurable_kernel f U mU).
apply/EFin_measurable_fun.
apply: (@measurable_comp _ _ _ _ _ _ [set r : R | r != 0%R]) => //.
+ exact: open_measurable.
+ move=> /= r [t] [] [_ ft0] ftoo ftr; apply/eqP => r0.
move: (ftr); rewrite r0 => /eqP; rewrite fine_eq0 ?ft0//.
by rewrite ge0_fin_numE// lt_neqAle leey ftoo.
+ apply: open_continuous_measurable_fun => //; apply/in_setP => x /= x0.
exact: inv_continuous.
+ apply: measurableT_comp => //=.
by have := measurable_kernel f _ measurableT; exact: measurable_funS.
Qed.
HB.instance Definition _ (P : probability Y R) :=
isKernel.Build _ _ _ _ R (knormalize P) (measurable_knormalize P).
Let knormalize1 (P : probability Y R) x : knormalize P x [set: Y] = 1.
Proof.
HB.instance Definition _ (P : probability Y R):=
@Kernel_isProbability.Build _ _ _ _ _ (knormalize P) (knormalize1 P).
End knormalize.
Lemma measurable_fun_mnormalize d d' (X : measurableType d)
(Y : measurableType d') (R : realType) (k : R.-ker X ~> Y) :
measurable_fun [set: X] (fun x =>
[the probability _ _ of mnormalize (k x) point] : pprobability Y R).
Proof.
apply: (@measurability _ _ _ _ _ _
(@pset _ _ _ : set (set (pprobability Y R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-.
rewrite /mnormalize /mset /preimage/=.
apply: emeasurable_fun_infty_o => //.
rewrite /mnormalize/=.
rewrite (_ : (fun x => _) = (fun x => if (k x setT == 0) || (k x setT == +oo)
then \d_point Ys else k x Ys * ((fine (k x setT))^-1)%:E)); last first.
by apply/funext => x/=; case: ifPn.
apply: measurable_fun_if => //.
- apply: (measurable_fun_bool true) => //.
rewrite (_ : _ @^-1` _ = [set t | k t setT == 0] `|`
[set t | k t setT == +oo]); last first.
by apply/seteqP; split=> x /= /orP.
by rewrite setTI; apply: measurableU; exact: kernel_measurable_eq_cst.
- apply/emeasurable_funM; first exact/measurable_funTS/measurable_kernel.
apply/EFin_measurable_fun; rewrite setTI.
apply: (@measurable_comp _ _ _ _ _ _ [set r : R | r != 0%R]).
+ exact: open_measurable.
+ by move=> /= _ [x /norP[s0 soo]] <-; rewrite -eqe fineK ?ge0_fin_numE ?ltey.
+ apply: open_continuous_measurable_fun => //; apply/in_setP => x /= x0.
exact: inv_continuous.
+ by apply: measurableT_comp => //; exact/measurable_funS/measurable_kernel.
Qed.
(@pset _ _ _ : set (set (pprobability Y R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-.
rewrite /mnormalize /mset /preimage/=.
apply: emeasurable_fun_infty_o => //.
rewrite /mnormalize/=.
rewrite (_ : (fun x => _) = (fun x => if (k x setT == 0) || (k x setT == +oo)
then \d_point Ys else k x Ys * ((fine (k x setT))^-1)%:E)); last first.
by apply/funext => x/=; case: ifPn.
apply: measurable_fun_if => //.
- apply: (measurable_fun_bool true) => //.
rewrite (_ : _ @^-1` _ = [set t | k t setT == 0] `|`
[set t | k t setT == +oo]); last first.
by apply/seteqP; split=> x /= /orP.
by rewrite setTI; apply: measurableU; exact: kernel_measurable_eq_cst.
- apply/emeasurable_funM; first exact/measurable_funTS/measurable_kernel.
apply/EFin_measurable_fun; rewrite setTI.
apply: (@measurable_comp _ _ _ _ _ _ [set r : R | r != 0%R]).
+ exact: open_measurable.
+ by move=> /= _ [x /norP[s0 soo]] <-; rewrite -eqe fineK ?ge0_fin_numE ?ltey.
+ apply: open_continuous_measurable_fun => //; apply/in_setP => x /= x0.
exact: inv_continuous.
+ by apply: measurableT_comp => //; exact/measurable_funS/measurable_kernel.
Qed.
Section kcomp_def.
Context d1 d2 d3 (X : measurableType d1) (Y : measurableType d2)
(Z : measurableType d3) (R : realType).
Variable l : X -> {measure set Y -> \bar R}.
Variable k : (X * Y)%type -> {measure set Z -> \bar R}.
Definition kcomp x U := \int[l x]_y k (x, y) U.
End kcomp_def.
Section kcomp_is_measure.
Context d1 d2 d3 (X : measurableType d1) (Y : measurableType d2)
(Z : measurableType d3) (R : realType).
Variable l : R.-ker X ~> Y.
Variable k : R.-ker [the measurableType _ of X * Y] ~> Z.
Let kcomp0 x : kcomp l k x set0 = 0.
Proof.
Let kcomp_ge0 x U : 0 <= kcomp l k x U. Proof. exact: integral_ge0.
Let kcomp_sigma_additive x : semi_sigma_additive (kcomp l k x).
Proof.
move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ =
\int[l x]_y (\sum_(n <oo) k (x, y) (U n))); last first.
apply: eq_integral => V _.
by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive.
apply/cvg_closeP; split.
by apply: is_cvg_nneseries => n _; exact: integral_ge0.
rewrite closeE// integral_nneseries// => n.
exact: measurableT_comp (measurable_kernel k _ (mU n)) _.
Qed.
\int[l x]_y (\sum_(n <oo) k (x, y) (U n))); last first.
apply: eq_integral => V _.
by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive.
apply/cvg_closeP; split.
by apply: is_cvg_nneseries => n _; exact: integral_ge0.
rewrite closeE// integral_nneseries// => n.
exact: measurableT_comp (measurable_kernel k _ (mU n)) _.
Qed.
HB.instance Definition _ x := isMeasure.Build _ _ R
(kcomp l k x) (kcomp0 x) (kcomp_ge0 x) (@kcomp_sigma_additive x).
Definition mkcomp : X -> {measure set Z -> \bar R} := fun x =>
[the measure _ _ of kcomp l k x].
End kcomp_is_measure.
Notation "l \; k" := (mkcomp l k) : ereal_scope.
Module KCOMP_FINITE_KERNEL.
Section kcomp_finite_kernel_kernel.
Context d d' d3 (X : measurableType d) (Y : measurableType d')
(Z : measurableType d3) (R : realType) (l : R.-fker X ~> Y)
(k : R.-ker [the measurableType _ of X * Y] ~> Z).
Lemma measurable_fun_kcomp_finite U :
measurable U -> measurable_fun [set: X] ((l \; k) ^~ U).
Proof.
move=> mU; apply: (measurable_fun_integral_finite_kernel (k ^~ U)) => //=.
exact/measurable_kernel.
Qed.
exact/measurable_kernel.
Qed.
HB.instance Definition _ :=
isKernel.Build _ _ X Z R (l \; k) measurable_fun_kcomp_finite.
End kcomp_finite_kernel_kernel.
Section kcomp_finite_kernel_finite.
Context d d' d3 (X : measurableType d) (Y : measurableType d')
(Z : measurableType d3) (R : realType).
Variable l : R.-fker X ~> Y.
Variable k : R.-fker [the measurableType _ of X * Y] ~> Z.
Let mkcomp_finite : measure_fam_uub (kcomp l k).
Proof.
have /measure_fam_uubP[r hr] := measure_uub k.
have /measure_fam_uubP[s hs] := measure_uub l.
apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=.
apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)).
apply: ge0_le_integral => //.
- exact: measurableT_comp (measurable_kernel k _ measurableT) _.
- by move=> y _; exact/ltW/hr.
by rewrite integral_cst//= EFinM lte_pmul2l.
Qed.
have /measure_fam_uubP[s hs] := measure_uub l.
apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=.
apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)).
apply: ge0_le_integral => //.
- exact: measurableT_comp (measurable_kernel k _ measurableT) _.
- by move=> y _; exact/ltW/hr.
by rewrite integral_cst//= EFinM lte_pmul2l.
Qed.
HB.instance Definition _ :=
Kernel_isFinite.Build _ _ X Z R (l \; k) mkcomp_finite.
End kcomp_finite_kernel_finite.
End KCOMP_FINITE_KERNEL.
Section kcomp_sfinite_kernel.
Context d d' d3 (X : measurableType d) (Y : measurableType d')
(Z : measurableType d3) (R : realType).
Variable l : R.-sfker X ~> Y.
Variable k : R.-sfker [the measurableType _ of X * Y] ~> Z.
Import KCOMP_FINITE_KERNEL.
Lemma mkcomp_sfinite : exists k_ : (R.-fker X ~> Z)^nat,
forall x U, measurable U -> (l \; k) x U = kseries k_ x U.
Proof.
have [k_ hk_] := sfinite_kernel k; have [l_ hl_] := sfinite_kernel l.
have [kl hkl] : exists kl : (R.-fker X ~> Z) ^nat, forall x U,
\esum_(i in setT) (l_ i.2 \; k_ i.1) x U = \sum_(i <oo) kl i x U.
have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card.
by rewrite card_eq_sym; exact: card_nat2.
exists (fun i => [the _.-fker _ ~> _ of l_ (f i).2 \; k_ (f i).1]) => x U.
by rewrite (reindex_esum [set: nat] _ f)// nneseries_esum// fun_true.
exists kl => x U mU.
transitivity (([the _.-ker _ ~> _ of kseries l_] \;
[the _.-ker _ ~> _ of kseries k_]) x U).
rewrite /= /kcomp [in RHS](eq_measure_integral (l x)); last first.
by move=> *; rewrite hl_.
by apply: eq_integral => y _; rewrite hk_.
rewrite /= /kcomp/= integral_nneseries//=; last first.
by move=> n; exact: measurableT_comp (measurable_kernel (k_ n) _ mU) _.
transitivity (\sum_(i <oo) \sum_(j <oo) (l_ j \; k_ i) x U).
apply: eq_eseriesr => i _; rewrite integral_kseries//.
by exact: measurableT_comp (measurable_kernel (k_ i) _ mU) _.
rewrite /mseries -hkl/=.
rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split.
rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//.
rewrite nneseries_esum; last by move=> n _; exact: nneseries_ge0.
by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true.
Qed.
have [kl hkl] : exists kl : (R.-fker X ~> Z) ^nat, forall x U,
\esum_(i in setT) (l_ i.2 \; k_ i.1) x U = \sum_(i <oo) kl i x U.
have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card.
by rewrite card_eq_sym; exact: card_nat2.
exists (fun i => [the _.-fker _ ~> _ of l_ (f i).2 \; k_ (f i).1]) => x U.
by rewrite (reindex_esum [set: nat] _ f)// nneseries_esum// fun_true.
exists kl => x U mU.
transitivity (([the _.-ker _ ~> _ of kseries l_] \;
[the _.-ker _ ~> _ of kseries k_]) x U).
rewrite /= /kcomp [in RHS](eq_measure_integral (l x)); last first.
by move=> *; rewrite hl_.
by apply: eq_integral => y _; rewrite hk_.
rewrite /= /kcomp/= integral_nneseries//=; last first.
by move=> n; exact: measurableT_comp (measurable_kernel (k_ n) _ mU) _.
transitivity (\sum_(i <oo) \sum_(j <oo) (l_ j \; k_ i) x U).
apply: eq_eseriesr => i _; rewrite integral_kseries//.
by exact: measurableT_comp (measurable_kernel (k_ i) _ mU) _.
rewrite /mseries -hkl/=.
rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split.
rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//.
rewrite nneseries_esum; last by move=> n _; exact: nneseries_ge0.
by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true.
Qed.
Lemma measurable_fun_mkcomp_sfinite U : measurable U ->
measurable_fun [set: X] ((l \; k) ^~ U).
Proof.
move=> mU; apply: (measurable_fun_integral_sfinite_kernel (k ^~ U)) => //.
exact/measurable_kernel.
Qed.
exact/measurable_kernel.
Qed.
End kcomp_sfinite_kernel.
Module KCOMP_SFINITE_KERNEL.
Section kcomp_sfinite_kernel.
Context d d' d3 (X : measurableType d) (Y : measurableType d')
(Z : measurableType d3) (R : realType).
Variable l : R.-sfker X ~> Y.
Variable k : R.-sfker [the measurableType _ of X * Y] ~> Z.
HB.instance Definition _ :=
isKernel.Build _ _ X Z R (l \; k) (measurable_fun_mkcomp_sfinite l k).
#[export]
HB.instance Definition _ :=
Kernel_isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k).
End kcomp_sfinite_kernel.
End KCOMP_SFINITE_KERNEL.
HB.export KCOMP_SFINITE_KERNEL.
Section measurable_fun_preimage_integral.
Context d d' (X : measurableType d) (Y : measurableType d') (R : realType).
Variables (k : Y -> \bar R)
(k_ : ({nnsfun Y >-> R}) ^nat)
(ndk_ : nondecreasing_seq (k_ : (Y -> R)^nat))
(k_k : forall z, [set: Y] z -> (k_ n z)%:E @[n --> \oo] --> k z).
Let k_2 : (X * Y -> R)^nat := fun n => k_ n \o snd.
Let k_2_ge0 n x : (0 <= k_2 n x)%R. Proof. by [].
HB.instance Definition _ n := @isNonNegFun.Build _ _ _ (k_2_ge0 n).
Let mk_2 n : measurable_fun [set: X * Y] (k_2 n).
Proof.
HB.instance Definition _ n := @isMeasurableFun.Build _ _ _ _ (mk_2 n).
Let fk_2 n : finite_set (range (k_2 n)).
Proof.
HB.instance Definition _ n := @FiniteImage.Build _ _ _ (fk_2 n).
Lemma measurable_fun_preimage_integral (l : X -> {measure set Y -> \bar R}) :
(forall n r, measurable_fun [set: X] (l ^~ (k_ n @^-1` [set r]))) ->
measurable_fun [set: X] (fun x => \int[l x]_z k z).
Proof.
move=> h; apply: (measurable_fun_xsection_integral (k \o snd) l
(fun n => [the {nnsfun _ >-> _} of k_2 n])) => /=.
- by rewrite /k_2 => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_.
- by move=> [x y]; exact: k_k.
- move=> n r _ /= B mB.
have := h n r measurableT B mB; rewrite !setTI.
suff : (l ^~ (k_ n @^-1` [set r])) @^-1` B =
(fun x => l x (xsection (k_2 n @^-1` [set r]) x)) @^-1` B by move=> ->.
by apply/seteqP; split => x/=;
rewrite (comp_preimage _ snd (k_ n)) xsection_preimage_snd.
Qed.
(fun n => [the {nnsfun _ >-> _} of k_2 n])) => /=.
- by rewrite /k_2 => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_.
- by move=> [x y]; exact: k_k.
- move=> n r _ /= B mB.
have := h n r measurableT B mB; rewrite !setTI.
suff : (l ^~ (k_ n @^-1` [set r])) @^-1` B =
(fun x => l x (xsection (k_2 n @^-1` [set r]) x)) @^-1` B by move=> ->.
by apply/seteqP; split => x/=;
rewrite (comp_preimage _ snd (k_ n)) xsection_preimage_snd.
Qed.
End measurable_fun_preimage_integral.
Lemma measurable_fun_integral_kernel
d d' (X : measurableType d) (Y : measurableType d') (R : realType)
(l : X -> {measure set Y -> \bar R})
(ml : forall U, measurable U -> measurable_fun [set: X] (l ^~ U))
(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.
Qed.
by apply: (measurable_fun_preimage_integral ndk_ k_k) => n r; exact/ml.
Qed.
Section integral_kcomp.
Context d d2 d3 (X : measurableType d) (Y : measurableType d2)
(Z : measurableType d3) (R : realType).
Variables (l : R.-sfker X ~> Y)
(k : R.-sfker [the measurableType _ of X * Y] ~> Z).
Let integral_kcomp_indic x E (mE : measurable E) :
\int[kcomp l k x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E).
Proof.
Let integral_kcomp_nnsfun x (f : {nnsfun Z >-> R}) :
\int[kcomp l k x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E).
Proof.
under [in LHS]eq_integral do rewrite fimfunE -fsumEFin//.
rewrite ge0_integral_fsum//; last 2 first.
- move=> r; apply/EFin_measurable_fun/measurableT_comp => [//|].
have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP.
by rewrite (_ : \1__ = mindic R fr).
- by move=> r z _; rewrite EFinM nnfun_muleindic_ge0.
under [in RHS]eq_integral.
move=> y _.
under eq_integral.
by move=> z _; rewrite fimfunE -fsumEFin//; over.
rewrite /= ge0_integral_fsum//; last 2 first.
- move=> r; apply/EFin_measurable_fun/measurableT_comp => [//|].
have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP.
by rewrite (_ : \1__ = mindic R fr).
- by move=> r z _; rewrite EFinM nnfun_muleindic_ge0.
under eq_fsbigr.
move=> r _.
rewrite (integralZl_indic _ (fun r => f @^-1` [set r]))//; last first.
by move=> r0; rewrite preimage_nnfun0.
rewrite integral_indic// setIT.
over.
over.
rewrite /= ge0_integral_fsum//; last 2 first.
- move=> r; apply: measurable_funeM.
exact: measurableT_comp (measurable_kernel k (f @^-1` [set r]) _) _.
- move=> n y _.
have := mulemu_ge0 (fun n => f @^-1` [set n]).
by apply; exact: preimage_nnfun0.
apply: eq_fsbigr => r _.
rewrite (integralZl_indic _ (fun r => f @^-1` [set r]))//; last first.
exact: preimage_nnfun0.
rewrite /= integral_kcomp_indic; last exact/measurable_sfunP.
have [r0|r0] := leP 0%R r.
rewrite ge0_integralZl//; last first.
exact: measurableT_comp (measurable_kernel k (f @^-1` [set r]) _) _.
by under eq_integral do rewrite integral_indic// setIT.
rewrite integral0_eq ?mule0; last first.
move=> y _; rewrite integral0_eq// => z _.
by rewrite preimage_nnfun0// indic0.
by rewrite integral0_eq// => y _; rewrite preimage_nnfun0// measure0 mule0.
Qed.
rewrite ge0_integral_fsum//; last 2 first.
- move=> r; apply/EFin_measurable_fun/measurableT_comp => [//|].
have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP.
by rewrite (_ : \1__ = mindic R fr).
- by move=> r z _; rewrite EFinM nnfun_muleindic_ge0.
under [in RHS]eq_integral.
move=> y _.
under eq_integral.
by move=> z _; rewrite fimfunE -fsumEFin//; over.
rewrite /= ge0_integral_fsum//; last 2 first.
- move=> r; apply/EFin_measurable_fun/measurableT_comp => [//|].
have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP.
by rewrite (_ : \1__ = mindic R fr).
- by move=> r z _; rewrite EFinM nnfun_muleindic_ge0.
under eq_fsbigr.
move=> r _.
rewrite (integralZl_indic _ (fun r => f @^-1` [set r]))//; last first.
by move=> r0; rewrite preimage_nnfun0.
rewrite integral_indic// setIT.
over.
over.
rewrite /= ge0_integral_fsum//; last 2 first.
- move=> r; apply: measurable_funeM.
exact: measurableT_comp (measurable_kernel k (f @^-1` [set r]) _) _.
- move=> n y _.
have := mulemu_ge0 (fun n => f @^-1` [set n]).
by apply; exact: preimage_nnfun0.
apply: eq_fsbigr => r _.
rewrite (integralZl_indic _ (fun r => f @^-1` [set r]))//; last first.
exact: preimage_nnfun0.
rewrite /= integral_kcomp_indic; last exact/measurable_sfunP.
have [r0|r0] := leP 0%R r.
rewrite ge0_integralZl//; last first.
exact: measurableT_comp (measurable_kernel k (f @^-1` [set r]) _) _.
by under eq_integral do rewrite integral_indic// setIT.
rewrite integral0_eq ?mule0; last first.
move=> y _; rewrite integral0_eq// => z _.
by rewrite preimage_nnfun0// indic0.
by rewrite integral0_eq// => y _; rewrite preimage_nnfun0// measure0 mule0.
Qed.
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).
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.
rewrite monotone_convergence//; last 3 first.
by move=> n; exact/EFin_measurable_fun.
by move=> n z _; rewrite lee_fin.
by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin.
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.
transitivity (\int[l x]_y lim ((\int[k (x, y)]_z (f_ n z)%:E) @[n --> \oo])).
rewrite -monotone_convergence//; last 3 first.
- move=> n; apply: measurable_fun_integral_kernel => //.
+ by move=> U mU; exact: measurableT_comp (measurable_kernel k _ mU) _.
+ by move=> z; rewrite lee_fin.
+ exact/EFin_measurable_fun.
- by move=> n y _; apply: integral_ge0 => // z _; rewrite lee_fin.
- move=> y _ a b ab; apply: ge0_le_integral => //.
+ by move=> z _; rewrite lee_fin.
+ exact/EFin_measurable_fun.
+ by move=> z _; rewrite lee_fin.
+ exact/EFin_measurable_fun.
+ by move: ab => /ndf_ /lefP ab z _; rewrite lee_fin.
apply: eq_integral => y _; rewrite -monotone_convergence//; last 3 first.
- by move=> n; exact/EFin_measurable_fun.
- 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.
Qed.
have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z).
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.
rewrite monotone_convergence//; last 3 first.
by move=> n; exact/EFin_measurable_fun.
by move=> n z _; rewrite lee_fin.
by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin.
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.
transitivity (\int[l x]_y lim ((\int[k (x, y)]_z (f_ n z)%:E) @[n --> \oo])).
rewrite -monotone_convergence//; last 3 first.
- move=> n; apply: measurable_fun_integral_kernel => //.
+ by move=> U mU; exact: measurableT_comp (measurable_kernel k _ mU) _.
+ by move=> z; rewrite lee_fin.
+ exact/EFin_measurable_fun.
- by move=> n y _; apply: integral_ge0 => // z _; rewrite lee_fin.
- move=> y _ a b ab; apply: ge0_le_integral => //.
+ by move=> z _; rewrite lee_fin.
+ exact/EFin_measurable_fun.
+ by move=> z _; rewrite lee_fin.
+ exact/EFin_measurable_fun.
+ by move: ab => /ndf_ /lefP ab z _; rewrite lee_fin.
apply: eq_integral => y _; rewrite -monotone_convergence//; last 3 first.
- by move=> n; exact/EFin_measurable_fun.
- 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.
Qed.
End integral_kcomp.