l3lab/ntp-mathlib huggingface dataset:
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : (fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊) =ᵐ[Measure.restrict μ s] 0
x : α
hx : ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ = OfNat.ofNat 0 x
⊢ ↑↑↑((condexpL2 ℝ ℝ hm) f) x = OfNat.ofNat 0 x
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0
⊢ Measurable fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊

Next tactic:
rw [Pi.zero_apply] at hx ⊢
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · 

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : (fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊) =ᵐ[Measure.restrict μ s] 0
x : α
hx : ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ = 0
⊢ ↑↑↑((condexpL2 ℝ ℝ hm) f) x = 0

Next tactic:
rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · 

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0
⊢ Measurable fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊

Next tactic:
refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0
⊢ Measurable fun x => ↑↑↑((condexpL2 ℝ ℝ hm) f) x

Next tactic:
rw [lpMeas_coe]
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
h_nnnorm_eq_zero : ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0
⊢ Measurable fun x => ↑↑↑((condexpL2 ℝ ℝ hm) f) x

Next tactic:
exact (Lp.stronglyMeasurable _).measurable
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
⊢ ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0

Next tactic:
refine' le_antisymm _ (zero_le _)
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
⊢ ∫⁻ (x : α) in s, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ ≤ 0

Next tactic:
refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
⊢ ∫⁻ (x : α) in s, ↑‖↑↑f x‖₊ ∂μ = 0

Next tactic:
rw [lintegral_eq_zero_iff]
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  rw [lintegral_eq_zero_iff]
  · 

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
⊢ (fun x => ↑‖↑↑f x‖₊) =ᵐ[Measure.restrict μ s] 0

Next tactic:
refine' hf.mono fun x hx => _
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  rw [lintegral_eq_zero_iff]
  · refine' hf.mono fun x hx => _
    

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
x : α
hx : ↑↑f x = OfNat.ofNat 0 x
⊢ (fun x => ↑‖↑↑f x‖₊) x = OfNat.ofNat 0 x

Next tactic:
dsimp only
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  rw [lintegral_eq_zero_iff]
  · refine' hf.mono fun x hx => _
    dsimp only
    

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
x : α
hx : ↑↑f x = OfNat.ofNat 0 x
⊢ ↑‖↑↑f x‖₊ = OfNat.ofNat 0 x

Next tactic:
rw [hx]
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  rw [lintegral_eq_zero_iff]
  · refine' hf.mono fun x hx => _
    dsimp only
    rw [hx]
    

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
x : α
hx : ↑↑f x = OfNat.ofNat 0 x
⊢ ↑‖OfNat.ofNat 0 x‖₊ = OfNat.ofNat 0 x

Next tactic:
simp
theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by
  suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0
  · rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
    refine' h_nnnorm_eq_zero.mono fun x hx => _
    dsimp only at hx
    rw [Pi.zero_apply] at hx ⊢
    · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
    · refine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)
      rw [lpMeas_coe]
      exact (Lp.stronglyMeasurable _).measurable
  refine' le_antisymm _ (zero_le _)
  refine' (lintegral_nnnorm_condexpL2_le hs hμs f).trans (le_of_eq _)
  rw [lintegral_eq_zero_iff]
  · refine' hf.mono fun x hx => _
    dsimp only
    rw [hx]
    simp
  · 

Proof state:
case h_nnnorm_eq_zero
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
hf : ↑↑f =ᵐ[Measure.restrict μ s] 0
⊢ Measurable fun x => ↑‖↑↑f x‖₊

Next tactic:
exact (Lp.stronglyMeasurable _).ennnorm
theorem lintegral_nnnorm_condexpL2_indicator_le_real (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    ∫⁻ a in t, ‖(condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ ≤ μ (s ∩ t) := by
  

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ ≤ ↑↑μ (s ∩ t)

Next tactic:
refine' (lintegral_nnnorm_condexpL2_le ht hμt _).trans (le_of_eq _)
theorem lintegral_nnnorm_condexpL2_indicator_le_real (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    ∫⁻ a in t, ‖(condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ ≤ μ (s ∩ t) := by
  refine' (lintegral_nnnorm_condexpL2_le ht hμt _).trans (le_of_eq _)
  

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ ∫⁻ (x : α) in t, ↑‖↑↑(indicatorConstLp 2 hs hμs 1) x‖₊ ∂μ = ↑↑μ (s ∩ t)

Next tactic:
have h_eq :
    ∫⁻ x in t, ‖(indicatorConstLp 2 hs hμs (1 : ℝ)) x‖₊ ∂μ =
      ∫⁻ x in t, s.indicator (fun _ => (1 : ℝ≥0∞)) x ∂μ := by
    refine' lintegral_congr_ae (ae_restrict_of_ae _)
    refine' (@indicatorConstLp_coeFn _ _ _ 2 _ _ _ hs hμs (1 : ℝ)).mono fun x hx => _
    dsimp only
    rw [hx]
    classical
    simp_rw [Set.indicator_apply]
    split_ifs <;> simp
theorem lintegral_nnnorm_condexpL2_indicator_le_real (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    ∫⁻ a in t, ‖(condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ ≤ μ (s ∩ t) := by
  refine' (lintegral_nnnorm_condexpL2_le ht hμt _).trans (le_of_eq _)
  have h_eq :
    ∫⁻ x in t, ‖(indicatorConstLp 2 hs hμs (1 : ℝ)) x‖₊ ∂μ =
      ∫⁻ x in t, s.indicator (fun _ => (1 : ℝ≥0∞)) x ∂μ := by
    

Proof state:
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
ht : MeasurableSet t
hμt : ↑↑μ t ≠ ⊤
⊢ ∫⁻ (x : α) in t, ↑‖↑↑(indicatorConstLp 2 hs hμs 1) x‖₊ ∂μ = ∫⁻ (x : α) in t, Set.indicator s (fun x => 1) x ∂μ

Next tactic:
refine' lintegral_congr_ae (ae_restrict_of_ae _)