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Merged
affeldt-aist merged 1 commit into from
Jun 26, 2025
Merged

Lfun1_integrable turned into an equivalence #1655

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20 changes: 11 additions & 9 deletions theories/hoelder.v
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Original file line number Diff line number Diff line change
Expand Up @@ -1088,16 +1088,18 @@ by under eq_integral => x _ do rewrite gee0_abs ?lee_fin ?powR_ge0//.
Qed.

Lemma Lfun1_integrable (f : T -> R) :
f \in Lfun mu 1 -> mu.-integrable setT (EFin \o f).
f \in Lfun mu 1 <-> mu.-integrable setT (EFin \o f).
Proof.
move=> /[dup] lf /Lfun_integrable => /(_ (lexx _)).
under eq_fun => x do rewrite powRr1//.
move/integrableP => [mf fley].
apply/integrableP; split.
move: lf; rewrite inE => /andP[/[!inE]/= {}mf _].
exact: measurableT_comp.
rewrite (le_lt_trans _ fley)//=.
by under [leRHS]eq_integral => x _ do rewrite normr_id.
split.
- move=> /[dup] lf /Lfun_integrable => /(_ (lexx _)).
under eq_fun => x do rewrite powRr1//.
move/integrableP => [mf fley]; apply/integrableP; split.
by move: lf; rewrite inE=> /andP[/[!inE]/= {}mf _]; exact: measurableT_comp.
rewrite (le_lt_trans _ fley)//=.
by under [leRHS]eq_integral => x _ do rewrite normr_id.
- move/integrableP => [mF iF].
rewrite inE; apply/andP; split; rewrite inE/=; first exact/measurable_EFinP.
by rewrite /finite_norm Lnorm1.
Qed.

Lemma Lfun2_integrable_sqr (f : T -> R) : f \in Lfun mu 2%:E ->
Expand Down
21 changes: 15 additions & 6 deletions theories/probability.v
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Expand Up @@ -261,7 +261,9 @@ Proof. by rewrite unlock. Qed.

Lemma expectation_fin_num (X : T -> R) : X \in Lfun P 1 ->
'E_P[X] \is a fin_num.
Proof. by move=> ?; rewrite unlock integral_fune_fin_num ?Lfun1_integrable. Qed.
Proof.
by move=> ?; rewrite unlock integral_fune_fin_num//; exact/Lfun1_integrable.
Qed.

Lemma expectation_cst r : 'E_P[cst r] = r%:E.
Proof. by rewrite unlock/= integral_cst//= probability_setT mule1. Qed.
Expand All @@ -278,7 +280,9 @@ Qed.

Lemma expectationZl (X : T -> R) (k : R) : X \in Lfun P 1 ->
'E_P[k \o* X] = k%:E * 'E_P [X].
Proof. by move=> ?; rewrite unlock muleC -integralZr ?Lfun1_integrable. Qed.
Proof.
by move=> ?; rewrite unlock muleC -integralZr//; exact/Lfun1_integrable.
Qed.

Lemma expectation_ge0 (X : T -> R) : (forall x, 0 <= X x)%R ->
0 <= 'E_P[X].
Expand All @@ -302,11 +306,15 @@ Qed.

Lemma expectationD (X Y : T -> R) : X \in Lfun P 1 -> Y \in Lfun P 1 ->
'E_P[X \+ Y] = 'E_P[X] + 'E_P[Y].
Proof. by move=> ? ?; rewrite unlock integralD_EFin ?Lfun1_integrable. Qed.
Proof.
by move=> ? ?; rewrite unlock integralD_EFin//; exact/Lfun1_integrable.
Qed.

Lemma expectationB (X Y : T -> R) : X \in Lfun P 1 -> Y \in Lfun P 1 ->
'E_P[X \- Y] = 'E_P[X] - 'E_P[Y].
Proof. by move=> ? ?; rewrite unlock integralB_EFin ?Lfun1_integrable. Qed.
Proof.
by move=> ? ?; rewrite unlock integralB_EFin//; exact/Lfun1_integrable.
Qed.

Lemma expectation_sum (X : seq (T -> R)) :
(forall Xi, Xi \in X -> Xi \in Lfun P 1) ->
Expand Down Expand Up @@ -710,11 +718,12 @@ have le (u : R) : (0 <= u)%R ->
- by rewrite lerD2r -lee_fin EFinB finEK.
apply: (le_trans (le_measure _ _ _ le)).
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
by apply: emeasurable_funB=> //; apply/measurable_int/(Lfun1_integrable X1).
apply: emeasurable_funB=> //.
by move/Lfun1_integrable : X1 => /measurable_int.
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
rewrite measurable_EFinP [X in measurable_fun _ X](_ : _ =
(fun x => x ^+ 2) \o (fun x => Y x + u))%R//.
by apply/measurableT_comp => //; apply/measurable_funD.
by apply/measurableT_comp => //; exact/measurable_funD.
set eps := ((lambda + u) ^ 2)%R.
have peps : (0 < eps)%R by rewrite exprz_gt0 ?ltr_wpDr.
rewrite (lee_pdivlMr _ _ peps) muleC.
Expand Down