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

integral_sum and fix integrable_sum #1650

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5 changes: 5 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -88,6 +88,8 @@

- in `set_interval.v`:
+ lemma `memB_itv`, `memB_itv0`
- in `lebesgue_integrable.v`:
+ lemma `integral_sum`

### Changed

Expand Down Expand Up @@ -237,6 +239,9 @@
- in `derive.v`:
+ lemmas `is_deriveX`, `deriveX`, `exp_derive`, `exp_derive1`

- in `lebesgue_integrable.v`:
+ lemma `integrable_sum`

### Deprecated

- in `set_interval.v`:
Expand Down
37 changes: 30 additions & 7 deletions theories/lebesgue_integral_theory/lebesgue_integrable.v
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Expand Up @@ -186,14 +186,15 @@ apply: (@le_lt_trans _ _ (\int[mu]_(x in D) (`|f x| + `|g x|))).
by rewrite ge0_integralD //; [exact: lte_add_pinfty| exact: measurableT_comp..].
Qed.

Lemma integrable_sum (s : seq (T -> \bar R)) :
(forall h, h \in s -> mu_int h) -> mu_int (fun x => \sum_(h <- s) h x).
Lemma integrable_sum I (s : seq I) (P : pred I) (h : I -> T -> \bar R) :
(forall i, P i -> mu_int (h i)) ->
mu_int (fun x => \sum_(i <- s | P i) h i x).
Proof.
elim: s => [_|h s ih hs].
elim: s => [_|i s ih hs].
by under eq_fun do rewrite big_nil; exact: integrable0.
under eq_fun do rewrite big_cons; apply: integrableD => //.
- by apply: hs; rewrite in_cons eqxx.
- by apply: ih => k ks; apply: hs; rewrite in_cons ks orbT.
under eq_fun do rewrite big_cons.
have [Pi|Pi] := boolP (P i); last exact: ih.
by apply: integrableD => //; [exact: hs|exact: ih].
Qed.

Lemma integrableB f g : mu_int f -> mu_int g -> mu_int (f \- g).
Expand Down Expand Up @@ -622,7 +623,7 @@ Section integralD.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType).
Variables (mu : {measure set T -> \bar R}) (D : set T) (mD : measurable D).
Variables (f1 f2 : T -> \bar R).
Variables f1 f2 : T -> \bar R.
Hypotheses (if1 : mu.-integrable D f1) (if2 : mu.-integrable D f2).

Let mf1 : measurable_fun D f1. Proof. exact: measurable_int if1. Qed.
Expand Down Expand Up @@ -681,6 +682,28 @@ Qed.

End integralD.

Section integral_sum.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType).
Variables (mu : {measure set T -> \bar R}) (D : set T) (mD : measurable D).
Variables (I : Type) (f : I -> (T -> \bar R)).
Hypothesis intf : forall n, mu.-integrable D (f n).

Lemma integral_sum (s : seq I) (P : pred I) :
\int[mu]_(x in D) (\sum_(k <- s | P k) f k x) =
\sum_(k <- s | P k) \int[mu]_(x in D) (f k x).
Proof.
elim: s => [|h t ih].
under eq_integral do rewrite big_nil.
by rewrite integral0 big_nil.
rewrite big_cons -ih -integralD//; last exact: integrable_sum.
case: ifPn => Ph.
by apply: eq_integral => x xD; rewrite big_cons Ph.
by apply: eq_integral => x xD; rewrite big_cons/= (negbTE Ph).
Qed.

End integral_sum.

Section integralB.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType).
Expand Down