Theorem: BoxIntegral.HasIntegral.of_aeEq_zero
File: Mathlib/Analysis/BoxIntegral/Integrability.lean

tactic: refine' hasIntegral_iff.2 fun ε ε0 => _
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
⊢ HasIntegral I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) 0

tactic: lift ε to ℝ≥0 using ε0.lt.le
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
ε : ℝ
ε0 : ε > 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ε

tactic: rw [gt_iff_lt, NNReal.coe_pos] at ε0
state_before: case intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : ↑ε > 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
state_before: case intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: haveI := Fact.mk (I.measure_coe_lt_top μ)
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: change μ.restrict I {x | f x ≠ 0} = 0 at hf
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[Measure.restrict μ ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: choose U hNU hUo hμU using this
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
this : ∀ (n : ℕ), ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ ↑↑(Measure.restrict μ ↑I) U < ↑(δ n) / ↑n
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: have : ∀ x, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U (N x) := fun x => by
  obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
  exact ⟨⟨r, hr₀⟩, hr⟩
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: choose r hrU using this
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
this : ∀ (x : ι → ℝ), ∃ r, closedBall x ↑r ⊆ U (N x)
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: refine' ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ _ => _⟩
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: rw [dist_eq_norm, sub_zero, ← integralSum_fiberwise fun J => N (π.tag J)]
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
⊢ dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

tactic: refine' le_trans _ (NNReal.coe_lt_coe.2 hcε).le
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
⊢ ‖∑ y in Finset.image (fun J => N (TaggedPrepartition.tag π J)) π.boxes,
        integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ))
          (TaggedPrepartition.filter π fun x => N (TaggedPrepartition.tag π x) = y)‖ ≤
    ↑ε

tactic: refine' (norm_sum_le_of_le _ _).trans
  (sum_le_hasSum _ (fun n _ => (δ n).2) (NNReal.hasSum_coe.2 hδc))
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
⊢ ‖∑ y in Finset.image (fun J => N (TaggedPrepartition.tag π J)) π.boxes,
        integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ))
          (TaggedPrepartition.filter π fun x => N (TaggedPrepartition.tag π x) = y)‖ ≤
    ↑c✝

tactic: rintro n -
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
⊢ ∀ (b : ℕ),
    b ∈ Finset.image (fun J => N (TaggedPrepartition.tag π J)) π.boxes →
      ‖integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ))
            (TaggedPrepartition.filter π fun x => N (TaggedPrepartition.tag π x) = b)‖ ≤
        ↑(δ b)

tactic: dsimp [integralSum]
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
⊢ ‖integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ))
        (TaggedPrepartition.filter π fun x => N (TaggedPrepartition.tag π x) = n)‖ ≤
    ↑(δ n)

tactic: have : ∀ J ∈ π.filter fun J => N (π.tag J) = n,
    ‖(μ ↑J).toReal • f (π.tag J)‖ ≤ (μ J).toReal * n := fun J hJ ↦ by
  rw [TaggedPrepartition.mem_filter] at hJ 
  rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
  exact mul_le_mul_of_nonneg_left (hJ.2 ▸ Nat.le_ceil _) ENNReal.toReal_nonneg
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
⊢ ‖∑ J in (TaggedPrepartition.filter π fun x => ⌈‖f (TaggedPrepartition.tag π x)‖⌉₊ = n).toPrepartition.boxes,
        ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤
    ↑(δ n)

tactic: refine' (norm_sum_le_of_le _ this).trans _
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
this :
  ∀ (J : Box ι),
    (J ∈ TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n) →
      ‖ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤ ENNReal.toReal (↑↑μ ↑J) * ↑n
⊢ ‖∑ J in (TaggedPrepartition.filter π fun x => ⌈‖f (TaggedPrepartition.tag π x)‖⌉₊ = n).toPrepartition.boxes,
        ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤
    ↑(δ n)

tactic: clear this
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
this :
  ∀ (J : Box ι),
    (J ∈ TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n) →
      ‖ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤ ENNReal.toReal (↑↑μ ↑J) * ↑n
⊢ ∑ b in (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n).toPrepartition.boxes,
      ENNReal.toReal (↑↑μ ↑b) * ↑n ≤
    ↑(δ n)

tactic: rw [← sum_mul, ← Prepartition.measure_iUnion_toReal]
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
⊢ ∑ b in (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n).toPrepartition.boxes,
      ENNReal.toReal (↑↑μ ↑b) * ↑n ≤
    ↑(δ n)

tactic: generalize hm : μ (π.filter fun J => N (π.tag J) = n).iUnion = m
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
⊢ ENNReal.toReal
        (↑↑μ
          (Prepartition.iUnion
            (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n).toPrepartition)) *
      ↑n ≤
    ↑(δ n)

tactic: have : m < δ n / n := by
  simp only [Measure.restrict_apply (hUo _).measurableSet] at hμU 
  refine' hm ▸ (measure_mono _).trans_lt (hμU _)
  simp only [Set.subset_def, TaggedPrepartition.mem_iUnion, TaggedPrepartition.mem_filter]
  rintro x ⟨J, ⟨hJ, rfl⟩, hx⟩
  exact ⟨hrU _ (hπ.1 _ hJ (Box.coe_subset_Icc hx)), π.le_of_mem' J hJ hx⟩
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
⊢ ENNReal.toReal m * ↑n ≤ ↑(δ n)

tactic: lift m to ℝ≥0 using ne_top_of_lt this
state_before: case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
this : m < ↑(δ n) / ↑n
⊢ ENNReal.toReal m * ↑n ≤ ↑(δ n)

tactic: rw [ENNReal.coe_toReal, ← NNReal.coe_nat_cast, ← NNReal.coe_mul, NNReal.coe_le_coe, ←
  ENNReal.coe_le_coe, ENNReal.coe_mul, ENNReal.coe_nat, mul_comm]
state_before: case intro.intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = ↑m
this : ↑m < ↑(δ n) / ↑n
⊢ ENNReal.toReal ↑m * ↑n ≤ ↑(δ n)

tactic: exact (mul_le_mul_left' this.le _).trans ENNReal.mul_div_le
state_before: case intro.intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = ↑m
this : ↑m < ↑(δ n) / ↑n
⊢ ↑n * ↑m ≤ ↑(δ n)

tactic: simp
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
⊢ ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0

tactic: refine' fun n => (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ _
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
⊢ ∀ (n : ℕ), ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ ↑↑(Measure.restrict μ ↑I) U < ↑(δ n) / ↑n

tactic: cases' n with n
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {n}) < ↑(δ n) / ↑n

tactic: simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
state_before: case zero
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {Nat.zero}) < ↑(δ Nat.zero) / ↑Nat.zero

tactic: refine' (measure_mono_null _ hf).le.trans_lt _
state_before: case succ
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {Nat.succ n}) < ↑(δ (Nat.succ n)) / ↑(Nat.succ n)

tactic: exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
state_before: case succ.refine'_1
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ N ⁻¹' {Nat.succ n} ⊆ {x | f x ≠ 0}

tactic: simp [(δ0 _).ne']
state_before: case succ.refine'_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ 0 < ↑(δ (Nat.succ n)) / ↑(Nat.succ n)

tactic: obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
x : ι → ℝ
⊢ ∃ r, closedBall x ↑r ⊆ U (N x)

tactic: exact ⟨⟨r, hr₀⟩, hr⟩
state_before: case intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
x : ι → ℝ
r : ℝ
hr₀ : 0 < r
hr : closedBall ?m.516813 r ⊆ U (N ?m.516813)
⊢ ∃ r, closedBall x ↑r ⊆ U (N x)

tactic: rw [TaggedPrepartition.mem_filter] at hJ
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
J : Box ι
hJ : J ∈ TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n
⊢ ‖ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤ ENNReal.toReal (↑↑μ ↑J) * ↑n

tactic: rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
J : Box ι
hJ : J ∈ π ∧ N (TaggedPrepartition.tag π J) = n
⊢ ‖ENNReal.toReal (↑↑μ ↑J) • f (TaggedPrepartition.tag π J)‖ ≤ ENNReal.toReal (↑↑μ ↑J) * ↑n

tactic: exact mul_le_mul_of_nonneg_left (hJ.2 ▸ Nat.le_ceil _) ENNReal.toReal_nonneg
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
J : Box ι
hJ : J ∈ π ∧ N (TaggedPrepartition.tag π J) = n
⊢ ENNReal.toReal (↑↑μ ↑J) * ‖f (TaggedPrepartition.tag π J)‖ ≤ ENNReal.toReal (↑↑μ ↑J) * ↑n

tactic: simp only [Measure.restrict_apply (hUo _).measurableSet] at hμU
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
⊢ m < ↑(δ n) / ↑n

tactic: refine' hm ▸ (measure_mono _).trans_lt (hμU _)
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
hμU : ∀ (n : ℕ), ↑↑μ (U n ∩ ↑I) < ↑(δ n) / ↑n
⊢ m < ↑(δ n) / ↑n

tactic: simp only [Set.subset_def, TaggedPrepartition.mem_iUnion, TaggedPrepartition.mem_filter]
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
hμU : ∀ (n : ℕ), ↑↑μ (U n ∩ ↑I) < ↑(δ n) / ↑n
⊢ TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n) ⊆ U n ∩ ↑I

tactic: rintro x ⟨J, ⟨hJ, rfl⟩, hx⟩
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
n : ℕ
m : ℝ≥0∞
hm : ↑↑μ (TaggedPrepartition.iUnion (TaggedPrepartition.filter π fun J => N (TaggedPrepartition.tag π J) = n)) = m
hμU : ∀ (n : ℕ), ↑↑μ (U n ∩ ↑I) < ↑(δ n) / ↑n
⊢ ∀ (x : ι → ℝ), (∃ J, (J ∈ π ∧ ⌈‖f (TaggedPrepartition.tag π J)‖⌉₊ = n) ∧ x ∈ J) → x ∈ U n ∩ ↑I

tactic: exact ⟨hrU _ (hπ.1 _ hJ (Box.coe_subset_Icc hx)), π.le_of_mem' J hJ hx⟩
state_before: case intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : MeasureTheory.Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : IntegrationParams.MemBaseSet l I c ((fun x => r) c) π
x✝ : TaggedPrepartition.IsPartition π
m : ℝ≥0∞
hμU : ∀ (n : ℕ), ↑↑μ (U n ∩ ↑I) < ↑(δ n) / ↑n
x : ι → ℝ
J : Box ι
hx : x ∈ J
hJ : J ∈ π
hm :
  ↑↑μ
      (TaggedPrepartition.iUnion
        (TaggedPrepartition.filter π fun J_1 =>
          N (TaggedPrepartition.tag π J_1) = ⌈‖f (TaggedPrepartition.tag π J)‖⌉₊)) =
    m
⊢ x ∈ U ⌈‖f (TaggedPrepartition.tag π J)‖⌉₊ ∩ ↑I