l3lab/ntp-mathlib huggingface dataset:
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  

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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
⊢ ∫ (a : α) in s, (↑↑↑((condexpL2 E' 𝕜 hm) f) - ↑↑f) a ∂μ = 0

Next tactic:
refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · 

Proof state:
case refine'_1
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
⊢ Integrable fun a => (↑↑↑((condexpL2 E' 𝕜 hm) f) - ↑↑f) a

Next tactic:
rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    

Proof state:
case refine'_1
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
⊢ Integrable fun x => ↑↑(↑((condexpL2 E' 𝕜 hm) f) - f) x

Next tactic:
exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
⊢ ∀ (c : E'), ∫ (x : α) in s, inner c ((↑↑↑((condexpL2 E' 𝕜 hm) f) - ↑↑f) x) ∂μ = 0

Next tactic:
intro c
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  intro c
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
c : E'
⊢ ∫ (x : α) in s, inner c ((↑↑↑((condexpL2 E' 𝕜 hm) f) - ↑↑f) x) ∂μ = 0

Next tactic:
simp_rw [Pi.sub_apply, inner_sub_right]
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  intro c
  simp_rw [Pi.sub_apply, inner_sub_right]
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
c : E'
⊢ ∫ (x : α) in s, inner c (↑↑↑((condexpL2 E' 𝕜 hm) f) x) - inner c (↑↑f x) ∂μ = 0

Next tactic:
rw [integral_sub
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)]
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  intro c
  simp_rw [Pi.sub_apply, inner_sub_right]
  rw [integral_sub
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)]
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
c : E'
⊢ ∫ (a : α) in s, inner c (↑↑↑((condexpL2 E' 𝕜 hm) f) a) ∂μ - ∫ (a : α) in s, inner c (↑↑f a) ∂μ = 0

Next tactic:
have h_ae_eq_f := Memℒp.coeFn_toLp (E := 𝕜) ((Lp.memℒp f).const_inner c)
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  intro c
  simp_rw [Pi.sub_apply, inner_sub_right]
  rw [integral_sub
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)]
  have h_ae_eq_f := Memℒp.coeFn_toLp (E := 𝕜) ((Lp.memℒp f).const_inner c)
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
c : E'
h_ae_eq_f :
  ↑↑(Memℒp.toLp (fun a => inner c (↑↑f a)) (_ : Memℒp (fun a => inner c (↑↑f a)) 2)) =ᵐ[μ] fun a => inner c (↑↑f a)
⊢ ∫ (a : α) in s, inner c (↑↑↑((condexpL2 E' 𝕜 hm) f) a) ∂μ - ∫ (a : α) in s, inner c (↑↑f a) ∂μ = 0

Next tactic:
rw [← lpMeas_coe, sub_eq_zero, ←
    set_integral_congr_ae (hm s hs) ((condexpL2_const_inner hm f c).mono fun x hx _ => hx), ←
    set_integral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]
/-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
  rw [← sub_eq_zero, lpMeas_coe, ←
    integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
      (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
  refine' integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ _ _
  · rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (↑(condexpL2 E' 𝕜 hm f)) f).symm)]
    exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs
  intro c
  simp_rw [Pi.sub_apply, inner_sub_right]
  rw [integral_sub
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)
      ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)]
  have h_ae_eq_f := Memℒp.coeFn_toLp (E := 𝕜) ((Lp.memℒp f).const_inner c)
  rw [← lpMeas_coe, sub_eq_zero, ←
    set_integral_congr_ae (hm s hs) ((condexpL2_const_inner hm f c).mono fun x hx _ => hx), ←
    set_integral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]
  

Proof state:
case refine'_2
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
f : ↥(Lp E' 2)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
c : E'
h_ae_eq_f :
  ↑↑(Memℒp.toLp (fun a => inner c (↑↑f a)) (_ : Memℒp (fun a => inner c (↑↑f a)) 2)) =ᵐ[μ] fun a => inner c (↑↑f a)
⊢ ∫ (x : α) in s,
      ↑↑↑((condexpL2 𝕜 𝕜 hm) (Memℒp.toLp (fun a => inner c (↑↑f a)) (_ : Memℒp (fun a => inner c (↑↑f a)) 2))) x ∂μ =
    ∫ (x : α) in s, ↑↑(Memℒp.toLp (fun a => inner c (↑↑f a)) (_ : Memℒp (fun a => inner c (↑↑f a)) 2)) x ∂μ

Next tactic:
exact integral_condexpL2_eq_of_fin_meas_real _ hs hμs
theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := 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✝¹⁸ : IsROrC 𝕜
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : InnerProductSpace 𝕜 E
inst✝¹⁵ : CompleteSpace E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : InnerProductSpace 𝕜 E'
inst✝¹² : CompleteSpace E'
inst✝¹¹ : NormedSpace ℝ E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : NormedSpace ℝ G'
inst✝⁵ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : IsROrC 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2)
⊢ ↑↑↑((condexpL2 E'' 𝕜' hm) (compLp T f)) =ᵐ[μ] ↑↑(compLp T ↑((condexpL2 E' 𝕜 hm) f))

Next tactic:
refine' Lp.ae_eq_of_forall_set_integral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
    (fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
      integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) _ _ _
theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := by
  refine' Lp.ae_eq_of_forall_set_integral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
    (fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
      integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) _ _ _
  · 

Proof state:
case refine'_1
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✝¹⁸ : IsROrC 𝕜
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : InnerProductSpace 𝕜 E
inst✝¹⁵ : CompleteSpace E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : InnerProductSpace 𝕜 E'
inst✝¹² : CompleteSpace E'
inst✝¹¹ : NormedSpace ℝ E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : NormedSpace ℝ G'
inst✝⁵ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : IsROrC 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2)
⊢ ∀ (s : Set α),
    MeasurableSet s →
      ↑↑μ s < ⊤ →
        ∫ (x : α) in s, ↑↑↑((condexpL2 E'' 𝕜' hm) (compLp T f)) x ∂μ =
          ∫ (x : α) in s, ↑↑(compLp T ↑((condexpL2 E' 𝕜 hm) f)) x ∂μ

Next tactic:
intro s hs hμs
theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := by
  refine' Lp.ae_eq_of_forall_set_integral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
    (fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
      integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) _ _ _
  · intro s hs hμs
    

Proof state:
case refine'_1
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✝¹⁸ : IsROrC 𝕜
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : InnerProductSpace 𝕜 E
inst✝¹⁵ : CompleteSpace E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : InnerProductSpace 𝕜 E'
inst✝¹² : CompleteSpace E'
inst✝¹¹ : NormedSpace ℝ E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : NormedSpace ℝ G'
inst✝⁵ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : IsROrC 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2)
s : Set α
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, ↑↑↑((condexpL2 E'' 𝕜' hm) (compLp T f)) x ∂μ =
    ∫ (x : α) in s, ↑↑(compLp T ↑((condexpL2 E' 𝕜 hm) f)) x ∂μ

Next tactic:
rw [T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne),
      ← lpMeas_coe, ← lpMeas_coe, integral_condexpL2_eq hm f hs hμs.ne,
      integral_condexpL2_eq hm (T.compLp f) hs hμs.ne, T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs.ne)]
theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := by
  refine' Lp.ae_eq_of_forall_set_integral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
    (fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
      integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) _ _ _
  · intro s hs hμs
    rw [T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne),
      ← lpMeas_coe, ← lpMeas_coe, integral_condexpL2_eq hm f hs hμs.ne,
      integral_condexpL2_eq hm (T.compLp f) hs hμs.ne, T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs.ne)]
  · 

Proof state:
case refine'_2
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✝¹⁸ : IsROrC 𝕜
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : InnerProductSpace 𝕜 E
inst✝¹⁵ : CompleteSpace E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : InnerProductSpace 𝕜 E'
inst✝¹² : CompleteSpace E'
inst✝¹¹ : NormedSpace ℝ E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : NormedSpace ℝ G'
inst✝⁵ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : IsROrC 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2)
⊢ AEStronglyMeasurable' m (↑↑↑((condexpL2 E'' 𝕜' hm) (compLp T f))) μ

Next tactic:
rw [← lpMeas_coe]
theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') := by
  refine' Lp.ae_eq_of_forall_set_integral_eq' 𝕜' hm _ _ two_ne_zero ENNReal.coe_ne_top
    (fun s _ hμs => integrableOn_condexpL2_of_measure_ne_top hm hμs.ne _) (fun s _ hμs =>
      integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne) _ _ _
  · intro s hs hμs
    rw [T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne),
      ← lpMeas_coe, ← lpMeas_coe, integral_condexpL2_eq hm f hs hμs.ne,
      integral_condexpL2_eq hm (T.compLp f) hs hμs.ne, T.set_integral_compLp _ (hm s hs),
      T.integral_comp_comm
        (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs.ne)]
  · rw [← lpMeas_coe]; 

Proof state:
case refine'_2
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✝¹⁸ : IsROrC 𝕜
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : InnerProductSpace 𝕜 E
inst✝¹⁵ : CompleteSpace E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : InnerProductSpace 𝕜 E'
inst✝¹² : CompleteSpace E'
inst✝¹¹ : NormedSpace ℝ E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : NormedSpace ℝ G'
inst✝⁵ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : IsROrC 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2)
⊢ AEStronglyMeasurable' m (↑↑↑((condexpL2 E'' 𝕜' hm) (compLp T f))) μ

Next tactic:
exact lpMeas.aeStronglyMeasurable' _