@@ -10,12 +10,13 @@ theorem Real.volume_uIoc {a b : ℝ} : volume (Set.uIoc a b) = ENNReal.ofReal |b
1010 /- Cf. proof of Real.volume_interval-/
1111 rw [Set.uIoc, volume_Ioc, max_sub_min_eq_abs]
1212
13- lemma intervalIntegral.integral_conj' {μ : Measure ℝ} {𝕜 : Type } [RCLike 𝕜] {f : ℝ → 𝕜} {a b : ℝ}:
13+ lemma intervalIntegral.integral_conj' {μ : Measure ℝ} {𝕜 : Type * } [RCLike 𝕜] {f : ℝ → 𝕜} {a b : ℝ} :
1414 ∫ x in a..b, (starRingEnd 𝕜) (f x) ∂μ = (starRingEnd 𝕜) (∫ x in a..b, f x ∂μ) := by
1515 rw [intervalIntegral_eq_integral_uIoc, integral_conj, intervalIntegral_eq_integral_uIoc,
1616 RCLike.real_smul_eq_coe_mul, RCLike.real_smul_eq_coe_mul, map_mul, RCLike.conj_ofReal]
1717
18- lemma intervalIntegrable_of_bdd {a b : ℝ} {δ : ℝ} {g : ℝ → ℂ} (measurable_g : Measurable g) (bddg : ∀ x, ‖g x‖ ≤ δ) : IntervalIntegrable g volume a b := by
18+ lemma intervalIntegrable_of_bdd {a b : ℝ} {δ : ℝ} {g : ℝ → ℂ} (measurable_g : Measurable g)
19+ (bddg : ∀ x, ‖g x‖ ≤ δ) : IntervalIntegrable g volume a b := by
1920 apply @IntervalIntegrable.mono_fun' _ _ _ _ _ _ (fun _ ↦ δ)
2021 · exact intervalIntegrable_const
2122 · exact measurable_g.aestronglyMeasurable
@@ -24,24 +25,26 @@ lemma intervalIntegrable_of_bdd {a b : ℝ} {δ : ℝ} {g : ℝ → ℂ} (measur
2425 rw [Subtype.forall]
2526 exact fun x _ ↦ bddg x
2627
27- lemma IntervalIntegrable.bdd_mul {F : Type } [NormedDivisionRing F] {f g : ℝ → F} {a b : ℝ} {μ : Measure ℝ}
28- (hg : IntervalIntegrable g μ a b) (hm : AEStronglyMeasurable f μ) (hfbdd : ∃ C, ∀ x, ‖f x‖ ≤ C) : IntervalIntegrable (fun x ↦ f x * g x) μ a b := by
28+ lemma IntervalIntegrable.bdd_mul {F : Type *} [NormedDivisionRing F] {f g : ℝ → F} {a b : ℝ}
29+ {μ : Measure ℝ} (hg : IntervalIntegrable g μ a b) (hm : AEStronglyMeasurable f μ)
30+ (hfbdd : ∃ C, ∀ x, ‖f x‖ ≤ C) : IntervalIntegrable (fun x ↦ f x * g x) μ a b := by
2931 rw [intervalIntegrable_iff, IntegrableOn]
3032 apply Integrable.bdd_mul _ hm.restrict hfbdd
3133 rwa [← IntegrableOn, ← intervalIntegrable_iff]
3234
33- lemma IntervalIntegrable.mul_bdd {F : Type } [NormedField F] {f g : ℝ → F} {a b : ℝ} {μ : Measure ℝ}
34- (hf : IntervalIntegrable f μ a b) (hm : AEStronglyMeasurable g μ) (hgbdd : ∃ C, ∀ x, ‖g x‖ ≤ C) : IntervalIntegrable (fun x ↦ f x * g x) μ a b := by
35+ lemma IntervalIntegrable.mul_bdd {F : Type *} [NormedField F] {f g : ℝ → F} {a b : ℝ}
36+ {μ : Measure ℝ} (hf : IntervalIntegrable f μ a b) (hm : AEStronglyMeasurable g μ)
37+ (hgbdd : ∃ C, ∀ x, ‖g x‖ ≤ C) : IntervalIntegrable (fun x ↦ f x * g x) μ a b := by
3538 conv => pattern (fun x ↦ f x * g x); ext x; rw [mul_comm]
3639 exact hf.bdd_mul hm hgbdd
3740
38- lemma IntegrableOn.sub {α : Type } {β : Type } {m : MeasurableSpace α}
39- {μ : Measure α} [NormedAddCommGroup β] {s : Set α} {f g : α → β} (hf : IntegrableOn f s μ) (hg : IntegrableOn g s μ) : IntegrableOn (f - g) s μ := by
41+ lemma IntegrableOn.sub {α : Type *} {β : Type *} {m : MeasurableSpace α} {μ : Measure α}
42+ [NormedAddCommGroup β] {s : Set α} {f g : α → β}
43+ (hf : IntegrableOn f s μ) (hg : IntegrableOn g s μ) : IntegrableOn (f - g) s μ := by
4044 apply Integrable.sub <;> rwa [← IntegrableOn]
4145
42-
43- lemma ConditionallyCompleteLattice.le_biSup {α : Type } [ConditionallyCompleteLinearOrder α] {ι : Type } [Nonempty ι]
44- {f : ι → α} {s : Set ι} {a : α} (hfs : BddAbove (f '' s)) (ha : ∃ i ∈ s, f i = a) :
46+ lemma ConditionallyCompleteLattice.le_biSup {α : Type *} [ConditionallyCompleteLinearOrder α]
47+ {ι : Type *} {f : ι → α} {s : Set ι} {a : α} (hfs : BddAbove (f '' s)) (ha : ∃ i ∈ s, f i = a) :
4548 a ≤ ⨆ i ∈ s, f i := by
4649 apply ConditionallyCompleteLattice.le_csSup
4750 · --TODO: improve this
@@ -75,15 +78,16 @@ lemma ConditionallyCompleteLattice.le_biSup {α : Type} [ConditionallyCompleteLi
7578 simp only [Set.mem_range, exists_prop] at hx
7679 rwa [hx.2 ] at fia
7780
78-
79- /-Adapted from mathlib Function.Periodic.exists_mem_Ico₀-/
80- theorem Function.Periodic.exists_mem_Ico₀' {α : Type } {β : Type } {f : α → β} {c : α}
81- [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c) (hc : 0 < c) (x : α) : ∃ (n : ℤ), (x - n • c) ∈ Set.Ico 0 c ∧ f x = f (x - n • c) :=
81+ /- Adapted from mathlib Function.Periodic.exists_mem_Ico₀ -/
82+ theorem Function.Periodic.exists_mem_Ico₀' {α β : Type *} {f : α → β} {c : α}
83+ [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c) (hc : 0 < c) (x : α) :
84+ ∃ (n : ℤ), (x - n • c) ∈ Set.Ico 0 c ∧ f x = f (x - n • c) :=
8285 let ⟨n, H, _⟩ := existsUnique_zsmul_near_of_pos' hc x
8386 ⟨n, H, (h.sub_zsmul_eq n).symm⟩
8487
85- /-Adapted from mathlib Function.Periodic.exists_mem_Ico₀-/
86- theorem Function.Periodic.exists_mem_Ico' {α : Type } {β : Type } {f : α → β} {c : α}
87- [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c) (hc : 0 < c) (x a: α) : ∃ (n : ℤ), (x - n • c) ∈ Set.Ico a (a + c) ∧ f x = f (x - n • c) :=
88+ /- Adapted from mathlib Function.Periodic.exists_mem_Ico₀ -/
89+ theorem Function.Periodic.exists_mem_Ico' {α β : Type *} {f : α → β} {c : α}
90+ [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c) (hc : 0 < c) (x a: α) :
91+ ∃ (n : ℤ), (x - n • c) ∈ Set.Ico a (a + c) ∧ f x = f (x - n • c) :=
8892 let ⟨n, H, _⟩ := existsUnique_sub_zsmul_mem_Ico hc x a
8993 ⟨n, H, (h.sub_zsmul_eq n).symm⟩
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