Module mathcomp.analysis.lebesgue_integral_theory.lebesgue_integral_monotone_convergence

From HB Require Import structures.
From mathcomp Require Import boot order ssralg ssrnum ssrint interval.
From mathcomp Require Import interval_inference archimedean finmap.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import boolp classical_sets functions cardinality fsbigop.
From mathcomp Require Import reals real_interval topology ereal tvs.
From mathcomp Require Import normedtype sequences esum measure.
From mathcomp Require Import lebesgue_measure numfun realfun simple_functions.
From mathcomp Require Import measurable_realfun measurable_fun_approximation.
From mathcomp Require Import lebesgue_integral_definition.

# Monotone convergence theorem for the Lebesgue Integral Monotone convergence and Fatou's lemmas for the Lebesgue integral.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldNormedType.Exports.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Section nondecreasing_integral_limit.
Local Open Scope ereal_scope.
Context ( : measurableType d) ( : realType).
Variables ( : {measure set T -> \bar R}) ( : T -> \bar R)
( : {nnsfun T >-> R}^nat).
Hypothesis : forall , 0 <= f x.
Hypothesis : measurable_fun setT f.

Import HBNNSimple.

Hypothesis : forall , nondecreasing_seq (g^~x).
Hypothesis : forall , EFin \o g^~ x @ \oo --> f x.
Local Open Scope ereal_scope.

Lemma : \int[mu]_ f x = limn (sintegral mu \o g).

Proof.

rewrite ge0_integralTE//.
apply/eqP; rewrite eq_le; apply/andP; split; last first.
  apply: lime_le; first exact: is_cvg_sintegral.
  near=> n; apply: ereal_sup_ubound; exists (g n) => //= => x.
  have <- : limn (EFin \o g ^~ x) = f x by exact/cvg_lim/gf.
  have : EFin \o g ^~ x @ \oo --> ereal_sup (range (EFin \o g ^~ x)).
    by apply: ereal_nondecreasing_cvgn => p q pq /=; rewrite lee_fin; exact/nd_g.
  by move/cvg_lim => -> //; apply: ereal_sup_ubound; exists n.
have := leey (\int[mu]_ (f x)).
rewrite [in X in X -> _]le_eqVlt => /predU1P[|] mufoo; last first.
  have : \int[mu]_ (f x) \is a fin_num by rewrite ge0_fin_numE// integral_ge0.
  rewrite ge0_integralTE// => /ub_ereal_sup_adherent h.
  apply/lee_addgt0Pr => _/posnumP[e].
  have {h} [/= _ [G Gf <-]] := h _ [gt0 of e%:num].
  rewrite EFinN lteBlDr// => fGe.
  have : forall , cvgn (g^~ x) -> (G x <= limn (g ^~ x))%R.
    move=> x cg; rewrite -lee_fin -(EFin_lim cg).
    by have /cvg_lim gxfx := @gf x; rewrite (le_trans (Gf _))// gxfx.
  move=> /(nd_sintegral_lim_lemma mu nd_g)/(leeD2r e%:num%:E).
  exact/le_trans/ltW.
suff : limn (sintegral mu \o g) = +oo.
  by move=> ->; rewrite -ge0_integralTE// mufoo.
apply/eqyP => r r0.
have [G [Gf rG]] : exists : {nnsfun T >-> R},
    (forall , (h x)%:E <= f x) /\ (r%:E <= sintegral mu h).
  have : r%:E < \int[mu]_ (f x).
    move: (mufoo) => /eqyP/(_ _ (addr_gt0 r0 r0)).
    by apply: lt_le_trans => //; rewrite lte_fin ltrDr.
  rewrite ge0_integralTE// => /ereal_sup_gt[x [/= G Gf Gx rx]].
  by exists G; split => //; rewrite (le_trans (ltW rx)) // Gx.
have : forall , cvgn (g^~ x) -> (G x <= limn (g^~ x))%R.
  move=> x cg; rewrite -lee_fin -(EFin_lim cg).
  by have /cvg_lim gxfx := @gf x; rewrite (le_trans (Gf _)) // gxfx.
by move/(nd_sintegral_lim_lemma mu nd_g) => Gg; rewrite (le_trans rG).
Unshelve. all: by end_near. Qed.

End nondecreasing_integral_limit.

Section ge0_le_integral.
Local Open Scope ereal_scope.
Context ( : measurableType d) ( : realType).
Variable : {measure set T -> \bar R}.
Variables ( : set T) ( : measurable D) ( : T -> \bar R).
Hypothesis : forall , D x -> 0 <= f1 x.
Hypothesis : measurable_fun D f1.
Hypothesis : measurable_fun D f2.

Import HBNNSimple.

Lemma : (forall , D x -> f1 x <= f2 x) ->
\int[mu]_( in D) f1 x <= \int[mu]_( in D) f2 x.

Proof.

move=> f12; rewrite !(integral_mkcond D).
set h1 := f1 \_ D; set h2 := f2 \_ D.
have h10 x : 0 <= h1 x by apply: erestrict_ge0.
have h20 x : 0 <= h2 x.
  by apply: erestrict_ge0 => // t /[dup] Dt /f12; apply: le_trans; exact: f10.
have mh1 : measurable_fun setT h1 by exact/(measurable_restrictT _ _).1.
have mh2 : measurable_fun setT h2 by exact/(measurable_restrictT _ _).1.
have h12 x : h1 x <= h2 x by apply: lee_restrict.
pose g1 := nnsfun_approx measurableT mh1.
rewrite (@nd_ge0_integral_lim _ _ _ _ _ g1)//.
  by move=> x m n mn; exact/lefP/nd_nnsfun_approx.
  by move=> x; exact: cvg_nnsfun_approx.
apply: lime_le.
  by apply: is_cvg_sintegral => // t m n mn; exact/lefP/nd_nnsfun_approx.
near=> n; rewrite ge0_integralTE//; apply: ereal_sup_ubound => /=.
exists (g1 n) => // t; rewrite (le_trans _ (h12 _))//.
have := leey (h1 t); rewrite le_eqVlt => /predU1P[->|ftoo].
  by rewrite leey.
have h1tfin : h1 t \is a fin_num.
  by rewrite fin_numE gt_eqF/= ?lt_eqF// (lt_le_trans _ (h10 t)).
have /= := @cvg_nnsfun_approx _ _ _ _ measurableT _ mh1 (fun _ => h10 x) t Logic.I.
rewrite -(fineK h1tfin) => /fine_cvgP[ft_near].
set u_ := (X in X --> _) => u_h1.
have <- : lim u_ = fine (h1 t) by exact/cvg_lim.
rewrite lee_fin; apply: nondecreasing_cvgn_le.
  by move=> // a b ab; rewrite /u_ /=; exact/lefP/nd_nnsfun_approx.
by apply/cvg_ex; eexists; exact: u_h1.
Unshelve. all: by end_near. Qed.

End ge0_le_integral.

Section monotone_convergence_theorem.
Local Open Scope ereal_scope.
Context ( : measurableType d) ( : realType).
Variable : {measure set T -> \bar R}.
Variables ( : set T) ( : measurable D) ( : (T -> \bar R)^nat).
Hypothesis : forall , measurable_fun D (g' n).
Hypothesis : forall , D x -> 0 <= g' n x.
Hypothesis : forall , D x -> nondecreasing_seq (g'^~ x).
Let := fun => limn (g'^~ x).

Let := (g' n \_ D).

Let : 0 <= g n x

Proof.

exact/erestrict_ge0/g'0. Qed.

Let : measurable_fun setT (g n).

Proof.

exact/(measurable_restrictT _ _).1. Qed.

Let : nondecreasing_seq (g^~ x).

Proof.

by move=> m n mn; rewrite /g/patch; case: ifP => // /set_mem /nd_g' ->.
Qed.

Let := fun => limn (g^~ x).

Let : cvgn (g^~ t).

Proof.

by move=> ?; apply: ereal_nondecreasing_is_cvgn => m n ?; exact/nd_g.
Qed.

Local Definition : (T -> R)^nat := approx setT (g n).
Local Definition : {nnsfun T >-> R}^nat := nnsfun_approx measurableT (mg n).

Local Definition : (T -> R)^nat :=
fun => (\big[maxr/0]_( < k) (g2' i k) t)%R.
Local Definition : {nnsfun T >-> R}^nat :=
fun => bigmax_nnsfun (g2^~ k) k.

Import HBNNSimple.

Let : cvgn (EFin \o (g2 n ^~ t)).

Proof.

apply: ereal_nondecreasing_is_cvgn => a b ab.
by rewrite lee_fin 2!nnsfun_approxE; exact/lefP/nd_approx.
Qed.

Let : nondecreasing_seq (max_g2 : (T -> R)^nat).

Proof.

apply/nondecreasing_seqP => n; apply/lefP => x; rewrite 2!bigmax_nnsfunE.
rewrite (@le_trans _ _ (\big[maxr/0]_( < n) g2 i n.+1 x)%R) //.
  apply: le_bigmax2 => i _; apply: (nondecreasing_seqP (g2 i ^~ x)).2 => a b ab.
   by rewrite !nnsfun_approxE; exact/lefP/nd_approx.
rewrite (bigmaxD1 ord_max)// le_max; apply/orP; right.
rewrite [leRHS](eq_bigl (fun => nat_of_ord i < n)%N); last first.
  by rewrite (big_ord_narrow (leqnSn n)).
move=> i /=; rewrite neq_lt; apply/orP/idP => [[//|]|]; last by left.
by move=> /(leq_trans (ltn_ord i)); rewrite ltnn.
Qed.

Let : cvgn (EFin \o max_g2 ^~ t).

Proof.

apply: ereal_nondecreasing_is_cvgn => m n mn; rewrite lee_fin.
exact/lefP/nd_max_g2.
Qed.

Let : ((max_g2 k x)%:E <= g k x)%O.

Proof.

rewrite bigmax_nnsfunE.
apply: (@le_trans _ _ (\big[maxe/0%:E]_( < k) g k x)); last first.
by apply/bigmax_leP; split => //; apply: g0D.
rewrite (big_morph _ (@EFin_max R) erefl) //.
apply: le_bigmax2 => i _; rewrite nnsfun_approxE /=.
by rewrite (le_trans (le_approx _ _ _)) => //; exact/nd_g/ltnW.
Qed.

Let : limn (EFin \o max_g2 ^~ t) <= f t.

Proof.

by apply: lee_lim => //=; near=> n; exact/max_g2_g.
Unshelve. all: by end_near. Qed.

Let : limn (EFin \o g2 n ^~ t) <= limn (EFin \o max_g2 ^~ t).

Proof.

apply: lee_lim => //.
near=> k; rewrite /= bigmax_nnsfunE lee_fin.
have nk : (n < k)%N by near: k; exists n.+1.
exact: (bigmax_sup (Ordinal nk)).
Unshelve. all: by end_near. Qed.

Let : EFin \o max_g2 ^~ t @ \oo --> f t.

Proof.

have /cvg_ex[l g_l] := @is_cvg_max_g2 t.
suff : l == f t by move=> /eqP <-.
rewrite eq_le; apply/andP; split.
  by rewrite /f (le_trans _ (lim_max_g2_f _)) // (cvg_lim _ g_l).
have := leey l; rewrite [in X in X -> _]le_eqVlt => /predU1P[->|loo].
  by rewrite leey.
rewrite -(cvg_lim _ g_l) //= lime_le => //.
near=> n.
have := leey (g n t); rewrite le_eqVlt => /predU1P[|] fntoo.
  have h := @dvg_approx _ _ _ setT _ t Logic.I fntoo.
  have g2oo : limn (EFin \o g2 n ^~ t) = +oo.
    apply/cvg_lim => //; apply/cvgeryP.
    under [in X in X --> _]eq_fun do rewrite nnsfun_approxE.
    have : {homo (approx setT (g n))^~ t : / (n0 <= m)%N >-> (n0 <= m)%R}.
      exact/lef_at/nd_approx.
    by move/nondecreasing_dvgn_lt => /(_ h).
  have -> : limn (EFin \o max_g2 ^~ t) = +oo.
    by have := lim_g2_max_g2 t n; rewrite g2oo leye_eq => /eqP.
  by rewrite leey.
- have approx_g_g := @cvg_approx _ _ _ setT _ t (fun _ => g0 n t) Logic.I fntoo.
  suff : limn (EFin \o g2 n ^~ t) = g n t.
    by move=> <-; exact: (le_trans _ (lim_g2_max_g2 t n)).
  have /cvg_lim <- // : EFin \o (approx setT (g n)) ^~ t @ \oo --> g n t.
    move/cvg_comp : approx_g_g; apply.
    by rewrite -(@fineK _ (g n t))// ge0_fin_numE// g0.
  rewrite (_ : _ \o _ = EFin \o approx setT (g n) ^~ t)// funeqE => m.
  by rewrite [in RHS]/= -nnsfun_approxE.
Unshelve. all: by end_near. Qed.

Lemma :
\int[mu]_( in D) (f' x) = limn (fun => \int[mu]_( in D) (g' n x)).

Proof.

rewrite integral_mkcond.
under [in RHS]eq_fun do rewrite integral_mkcond -/(g _).
have -> : f' \_ D = f.
  apply/funext => x; rewrite /f /f' /g /patch /=; case: ifPn => //=.
  by rewrite lim_cst.
apply/eqP; rewrite eq_le; apply/andP; split; last first.
  have nd_int_g : nondecreasing_seq (fun => \int[mu]_ g n x).
    move=> m n mn; apply: ge0_le_integral => //.
    by move=> *; exact: nd_g.
  have ub n : \int[mu]_ g n x <= \int[mu]_ f x.
    apply: ge0_le_integral => //.
    - apply: emeasurable_fun_cvg mg _ => x _.
      exact: ereal_nondecreasing_is_cvgn.
    - move=> x Dx; apply: lime_ge => //.
      near=> m; have nm : (n <= m)%N by near: m; exists n.
      exact/nd_g.
  by apply: lime_le => //; [exact:ereal_nondecreasing_is_cvgn|exact:nearW].
rewrite (@nd_ge0_integral_lim _ _ _ mu _ max_g2) //.
  - move=> t; apply: lime_ge => //.
    by apply: nearW => n; exact: g0.
  - by move=> t m n mn; exact/lefP/nd_max_g2.
apply: lee_lim.
- by apply: is_cvg_sintegral => // t m n mn; exact/lefP/nd_max_g2.
- apply: ereal_nondecreasing_is_cvgn => // n m nm; apply: ge0_le_integral => //.
  by move=> *; exact/nd_g.
- apply: nearW => n; rewrite ge0_integralTE//.
  by apply: ereal_sup_ubound; exists (max_g2 n) => // t; exact: max_g2_g.
Unshelve. all: by end_near. Qed.

Lemma :
\int[mu]_( in D) g' n x @[ \oo] --> \int[mu]_( in D) f' x.

Proof.

rewrite monotone_convergence; apply: ereal_nondecreasing_is_cvgn => m n mn.
by apply: ge0_le_integral => // t Dt; [exact: g'0|exact: nd_g'].
Qed.

End monotone_convergence_theorem.

Section fatou.
Local Open Scope ereal_scope.
Context ( : measurableType d) ( : realType).
Variables ( : {measure set T -> \bar R}) ( : set T) ( : measurable D).
Variable : (T -> \bar R)^nat.
Hypothesis : forall , measurable_fun D (f n).
Hypothesis : forall , D x -> 0 <= f n x.

Lemma : \int[mu]_( in D) limn_einf (f^~ x) <=
limn_einf (fun => \int[mu]_( in D) f n x).

Proof.

pose g := fun => einfs (f ^~ x) n.
have mg := measurable_fun_einfs mf.
have g0 n x : D x -> 0 <= g n x.
  by move=> Dx; apply: le_ereal_inf_tmp => _ [m /= nm <-]; exact: f0.
under eq_integral do rewrite limn_einf_lim.
rewrite limn_einf_lim monotone_convergence //.
  move=> x Dx m n mn /=; apply: ereal_inf_le_tmp => _ /= [p /= np <-].
  by exists p => //=; rewrite (leq_trans mn).
apply: lee_lim.
- apply/cvg_ex; eexists; apply/ereal_nondecreasing_cvgn => a b ab.
  apply: ge0_le_integral => //; [exact: g0| exact: mg| exact: mg|].
  move=> x Dx; apply: ereal_inf_le_tmp => _ [n /= bn <-].
  by exists n => //=; rewrite (leq_trans ab).
- apply/cvg_ex; eexists; apply/ereal_nondecreasing_cvgn => a b ab.
  apply: ereal_inf_le_tmp => // _ [n /= bn <-].
  by exists n => //=; rewrite (leq_trans ab).
- apply: nearW => m.
  have : forall , (p >= n)%N ->
      \int[mu]_( in D) g n x <= einfs (fun => \int[mu]_( in D) f k x) n.
    move=> n p np; apply: le_ereal_inf_tmp => /= _ [k /= nk <-].
    apply: ge0_le_integral => //; [exact: g0|exact: mg|].
    by move=> x Dx; apply: ereal_inf_lbound; exists k.
  exact.
Qed.

End fatou.

Files

Module mathcomp.analysis.lebesgue_integral_theory.lebesgue_integral_monotone_convergence