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fixes #1299 (minor generalization) #1302

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Aug 24, 2024
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fixes #1299 (minor generalization) #1302

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -63,6 +63,9 @@
- in `lebesgue_measure.v`:
+ lemma `measurable_funX` (was `measurable_fun_pow`)

- in `lebesgue_integral.v`
+ lemma `ge0_integral_closed_ball`

### Deprecated

### Removed
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48 changes: 22 additions & 26 deletions theories/lebesgue_integral.v
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Expand Up @@ -6358,26 +6358,27 @@ Qed.
Lemma integral_set1 (f : R -> \bar R) (r : R) : \int[mu]_(x in [set r]) f x = 0.
Proof. exact: (integral_Sset1 _ (@subset_refl _ _)). Qed.

Lemma ge0_integral_closed_ball (r : R) (r0 : (0 < r)%R) (f : R -> \bar R) :
measurable_fun (closed_ball 0%R r : set R^o) f ->
(forall x, x \in closed_ball 0%R r -> 0 <= f x) ->
\int[mu]_(x in closed_ball 0%R r) f x = \int[mu]_(x in ball 0%R r) f x.
Lemma ge0_integral_closed_ball c (r : R) (r0 : (0 < r)%R) (f : R -> \bar R) :
measurable_fun (closed_ball c r : set R) f ->
(forall x, x \in closed_ball c r -> 0 <= f x) ->
\int[mu]_(x in closed_ball c r) f x = \int[mu]_(x in ball c r) f x.
Proof.
move=> mf f0.
rewrite closed_ball_ball//= sub0r add0r ge0_integral_setU//=; last 4 first.
rewrite closed_ball_ball//= ge0_integral_setU//=; last 4 first.
by apply: measurableU => //; exact: measurable_ball.
by move: mf; rewrite closed_ball_ball// sub0r add0r.
by move=> x H; rewrite f0// closed_ball_ball// inE/= sub0r add0r.
apply/disj_setPLR => x [->/=|]; first by apply/eqP; rewrite lt_eqF// gtrN.
by rewrite /ball/= sub0r normrN => /ltr_normlW ?; apply/eqP; rewrite lt_eqF.
rewrite integral_set1 adde0 ge0_integral_setU//=.
- by rewrite integral_set1//= ?add0e.
by move: mf; rewrite closed_ball_ball.
by move=> x xcr; rewrite f0// closed_ball_ball// inE.
apply/disj_setPLR => x [->|]/=; rewrite /ball/=.
by apply/eqP; rewrite (addrC _ r) -subr_eq -addrA addrC subrK eqNr gt_eqF.
by move=> /[swap] ->; rewrite opprD addrA subrr sub0r normrN gtr0_norm// ltxx.
rewrite ge0_integral_setU//=.
- by rewrite !integral_set1//= add0e adde0.
- exact: measurable_ball.
- apply: measurable_funS mf; first exact: measurable_closed_ball.
by move=> x; rewrite closed_ball_ball// sub0r add0r; left.
- by move=> x H; rewrite f0// closed_ball_ball// inE/= sub0r add0r; left.
- apply/disj_setPRL => x/=; rewrite /ball/= sub0r => /ltr_normlW ?.
by apply/eqP; rewrite gt_eqF// ltrNl.
by move=> x; rewrite closed_ball_ball//; left.
- by move=> x H; rewrite f0// closed_ball_ball// inE/=; left.
- apply/disj_setPRL => x /[swap] ->.
by rewrite /ball/= opprB addrCA subrr addr0 gtr0_norm// ltxx.
Qed.

Lemma integral_setD1_EFin (f : R -> R) r (D : set R) :
Expand Down Expand Up @@ -6687,18 +6688,15 @@ have fE y k r : (ball 0%R k.+1%:R) y -> (r < 1)%R ->
have [t0|t0] := leP 0%R t.
rewrite ger0_norm// (lt_le_trans txr)// -lerBrDr.
rewrite (le_trans (ler_norm _))// (le_trans (ltW yk1))// -lerBlDr.
rewrite opprK -lerBrDl -natrB//; last by rewrite -addnn addSnnS ltn_addl.
by rewrite (le_trans (ltW r1))// -addnn addnK// ler1n.
by rewrite opprK -lerBrDl -addnn natrD addrK (le_trans (ltW r1))// ler1n.
rewrite -opprB ltrNl in xrt.
rewrite ltr0_norm// (lt_le_trans xrt)// lerBlDl (le_trans (ltW r1))//.
rewrite -lerBlDl addrC -lerBrDr (le_trans (ler_norm _))//.
rewrite -normrN in yk1.
rewrite (le_trans (ltW yk1))// -lerBlDr opprK -lerBrDl.
rewrite -natrB//; last by rewrite -addnn addSnnS ltn_addl.
by rewrite -addnn addnK ler1n.
by rewrite (le_trans (ltW yk1))// lerBrDr natr1 ler_nat -muln2 ltn_Pmulr.
have := h `|ceil x|.+1%N Logic.I.
have Bxx : B `|ceil x|.+1 x.
rewrite /B /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))// ltr_nat.
rewrite /B /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))// ltr_nat.
by rewrite -addnn addSnnS ltn_addl.
move=> /(_ Bxx)/fine_cvgP[davg_fk_fin_num davg_fk0].
have f_fk_ceil : \forall t \near 0^'+,
Expand All @@ -6707,15 +6705,13 @@ have f_fk_ceil : \forall t \near 0^'+,
near=> t.
apply: eq_integral => /= y /[1!inE] xty.
rewrite -(fE x _ t)//; last first.
rewrite /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))//.
by rewrite -[in ltRHS]natr1 ltrDl.
by rewrite /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))// ltr_nat.
rewrite -(fE x _ t)//; last first.
by apply: ballxx; near: t; exact: nbhs_right_gt.
rewrite /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))//.
by rewrite -[in ltRHS]natr1 ltrDl.
by rewrite /ball/= sub0r normrN (le_lt_trans (abs_ceil_ge _))// ltr_nat.
apply/fine_cvgP; split=> [{davg_fk0}|{davg_fk_fin_num}].
- move: davg_fk_fin_num => -[M /= M0] davg_fk_fin_num.
apply: filter_app f_fk_ceil; near=> t; move=> Ht.
apply: filter_app f_fk_ceil; near=> t => Ht.
by rewrite /davg /iavg Ht davg_fk_fin_num//= sub0r normrN gtr0_norm.
- move/cvgrPdist_le in davg_fk0; apply/cvgrPdist_le => e e0.
have [M /= M0 {}davg_fk0] := davg_fk0 _ e0.
Expand Down
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