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fixes #1764 (continuity under) #1765

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Merged
affeldt-aist merged 1 commit into from
Nov 19, 2025
Merged

fixes #1764 (continuity under) #1765

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -13,6 +13,9 @@

### Generalized

- in `lebesgue_integral_under.v`:
+ weaken an hypothesis of lemma `continuity_under_integral`

### Deprecated

### Removed
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15 changes: 8 additions & 7 deletions theories/lebesgue_integral_theory/lebesgue_integral_under.v
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Expand Up @@ -41,12 +41,11 @@ Variable f : R -> Y -> R.
Variable B : set Y.
Hypothesis mB : measurable B.

Variable a u v : R.
Variable u v : R.
Let I : set R := `]u, v[.

Hypothesis Ia : I a.
Hypothesis int_f : forall x, I x -> mu.-integrable B (EFin \o (f x)).
Hypothesis cf : {ae mu, forall y, B y -> continuous (f ^~ y)}.
Hypothesis cf : {ae mu, forall y, B y -> {in I, continuous (f ^~ y)}}.

Variable g : Y -> R.

Expand All @@ -56,13 +55,14 @@ Hypothesis g_ub : forall x, I x -> {ae mu, forall y, B y -> `|f x y| <= g y}.
Let F x := (\int[mu]_(y in B) f x y)%R.

Lemma continuity_under_integral :
continuous_at a (fun l => \int[mu]_(x in B) f l x).
{in I, continuous (fun l => \int[mu]_(x in B) f l x)}.
Proof.
move=> a /set_mem Ia.
have [Z [mZ Z0 /subsetCPl ncfZ]] := cf.
have BZ_cf x : x \in B `\` Z -> continuous (f ^~ x).
have BZ_cf x : x \in B `\` Z -> {in I, continuous (f ^~ x)}.
by rewrite inE/= => -[Bx nZx]; exact: ncfZ.
have [vu|uv] := lerP v u.
by move: Ia; rewrite /I set_itv_ge// -leNgt bnd_simp.
by move: (Ia); rewrite /I set_itv_ge// -leNgt bnd_simp.
apply/cvg_nbhsP => w wa.
have /near_in_itvoo[e /= e0 aeuv] : a \in `]u, v[ by rewrite inE.
move/cvgrPdist_lt : (wa) => /(_ _ e0)[N _ aue].
Expand Down Expand Up @@ -114,7 +114,8 @@ apply: (@dominated_cvg _ _ _ mu _ _
have : x \in B `\` Z.
move: BZUUx; rewrite inE/= => -[Bx nZUUx]; rewrite inE/=; split => //.
by apply: contra_not nZUUx; left.
by move/(BZ_cf x)/(_ a)/cvg_nbhsP; apply; rewrite (cvg_shiftn N).
move/(BZ_cf x)/(_ a); move/mem_set : Ia => /[swap] /[apply].
by move/cvg_nbhsP; apply; rewrite (cvg_shiftn N).
- by apply: (integrableS mB) => //; exact: measurableD.
- move=> n x [Bx ZUUx]; rewrite lee_fin.
move/subsetCPl : (Ug_ub n); apply => //=.
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