l3lab/ntp-mathlib huggingface dataset:
theorem condexpL1Clm_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condexpL1Clm F' hm μ (f : α →₁[μ] F') = ↑f := by
  let g := lpMeasToLpTrimLie F' ℝ 1 μ hm f
  have hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g := by
    simp only [LinearIsometryEquiv.symm_apply_apply]
  rw [hfg]
  refine' @Lp.induction α F' m _ 1 (μ.trim hm) _ ENNReal.coe_ne_top (fun g : α →₁[μ.trim hm] F' =>
    condexpL1Clm F' hm μ ((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g : α →₁[μ] F') =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g)) _ _ _ g
  · intro c s hs hμs
    rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs hμs.ne c,
      condexpL1Clm_indicatorConstLp]
    exact condexpInd_of_measurable hs ((le_trim hm).trans_lt hμs).ne c
  · intro f g hf hg _ hf_eq hg_eq
    rw [LinearIsometryEquiv.map_add]
    push_cast
    rw [map_add, hf_eq, hg_eq]
  · refine' isClosed_eq _ _
    · 

Proof state:
case refine'_3.refine'_1
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) g
⊢ Continuous fun f => (condexpL1Clm F' hm μ) ↑((LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) f)

Next tactic:
refine' (condexpL1Clm F' hm μ).continuous.comp (continuous_induced_dom.comp _)
theorem condexpL1Clm_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condexpL1Clm F' hm μ (f : α →₁[μ] F') = ↑f := by
  let g := lpMeasToLpTrimLie F' ℝ 1 μ hm f
  have hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g := by
    simp only [LinearIsometryEquiv.symm_apply_apply]
  rw [hfg]
  refine' @Lp.induction α F' m _ 1 (μ.trim hm) _ ENNReal.coe_ne_top (fun g : α →₁[μ.trim hm] F' =>
    condexpL1Clm F' hm μ ((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g : α →₁[μ] F') =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g)) _ _ _ g
  · intro c s hs hμs
    rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs hμs.ne c,
      condexpL1Clm_indicatorConstLp]
    exact condexpInd_of_measurable hs ((le_trim hm).trans_lt hμs).ne c
  · intro f g hf hg _ hf_eq hg_eq
    rw [LinearIsometryEquiv.map_add]
    push_cast
    rw [map_add, hf_eq, hg_eq]
  · refine' isClosed_eq _ _
    · refine' (condexpL1Clm F' hm μ).continuous.comp (continuous_induced_dom.comp _)
      

Proof state:
case refine'_3.refine'_1
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) g
⊢ Continuous fun f => (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) f

Next tactic:
exact LinearIsometryEquiv.continuous _
theorem condexpL1Clm_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condexpL1Clm F' hm μ (f : α →₁[μ] F') = ↑f := by
  let g := lpMeasToLpTrimLie F' ℝ 1 μ hm f
  have hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g := by
    simp only [LinearIsometryEquiv.symm_apply_apply]
  rw [hfg]
  refine' @Lp.induction α F' m _ 1 (μ.trim hm) _ ENNReal.coe_ne_top (fun g : α →₁[μ.trim hm] F' =>
    condexpL1Clm F' hm μ ((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g : α →₁[μ] F') =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g)) _ _ _ g
  · intro c s hs hμs
    rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs hμs.ne c,
      condexpL1Clm_indicatorConstLp]
    exact condexpInd_of_measurable hs ((le_trim hm).trans_lt hμs).ne c
  · intro f g hf hg _ hf_eq hg_eq
    rw [LinearIsometryEquiv.map_add]
    push_cast
    rw [map_add, hf_eq, hg_eq]
  · refine' isClosed_eq _ _
    · refine' (condexpL1Clm F' hm μ).continuous.comp (continuous_induced_dom.comp _)
      exact LinearIsometryEquiv.continuous _
    · 

Proof state:
case refine'_3.refine'_2
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) g
⊢ Continuous fun f => ↑((LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) f)

Next tactic:
refine' continuous_induced_dom.comp _
theorem condexpL1Clm_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condexpL1Clm F' hm μ (f : α →₁[μ] F') = ↑f := by
  let g := lpMeasToLpTrimLie F' ℝ 1 μ hm f
  have hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g := by
    simp only [LinearIsometryEquiv.symm_apply_apply]
  rw [hfg]
  refine' @Lp.induction α F' m _ 1 (μ.trim hm) _ ENNReal.coe_ne_top (fun g : α →₁[μ.trim hm] F' =>
    condexpL1Clm F' hm μ ((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g : α →₁[μ] F') =
    ↑((lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g)) _ _ _ g
  · intro c s hs hμs
    rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs hμs.ne c,
      condexpL1Clm_indicatorConstLp]
    exact condexpInd_of_measurable hs ((le_trim hm).trans_lt hμs).ne c
  · intro f g hf hg _ hf_eq hg_eq
    rw [LinearIsometryEquiv.map_add]
    push_cast
    rw [map_add, hf_eq, hg_eq]
  · refine' isClosed_eq _ _
    · refine' (condexpL1Clm F' hm μ).continuous.comp (continuous_induced_dom.comp _)
      exact LinearIsometryEquiv.continuous _
    · refine' continuous_induced_dom.comp _
      

Proof state:
case refine'_3.refine'_2
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) g
⊢ Continuous fun f => (LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' ℝ 1 μ hm)) f

Next tactic:
exact LinearIsometryEquiv.continuous _
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
⊢ AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ

Next tactic:
by_cases hf : Integrable f μ
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  by_cases hf : Integrable f μ
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
hf : Integrable f
⊢ AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ

Next tactic:
rw [condexpL1_eq hf]
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  by_cases hf : Integrable f μ
  · rw [condexpL1_eq hf]
    

Proof state:
case pos
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
hf : Integrable f
⊢ AEStronglyMeasurable' m (↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hf))) μ

Next tactic:
exact aestronglyMeasurable'_condexpL1Clm _
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  by_cases hf : Integrable f μ
  · rw [condexpL1_eq hf]
    exact aestronglyMeasurable'_condexpL1Clm _
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
hf : ¬Integrable f
⊢ AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ

Next tactic:
rw [condexpL1_undef hf]
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  by_cases hf : Integrable f μ
  · rw [condexpL1_eq hf]
    exact aestronglyMeasurable'_condexpL1Clm _
  · rw [condexpL1_undef hf]
    

Proof state:
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
hf : ¬Integrable f
⊢ AEStronglyMeasurable' m (↑↑0) μ

Next tactic:
refine AEStronglyMeasurable'.congr ?_ (coeFn_zero _ _ _).symm
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ := by
  by_cases hf : Integrable f μ
  · rw [condexpL1_eq hf]
    exact aestronglyMeasurable'_condexpL1Clm _
  · rw [condexpL1_undef hf]
    refine AEStronglyMeasurable'.congr ?_ (coeFn_zero _ _ _).symm
    

Proof state:
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
f : α → F'
hf : ¬Integrable f
⊢ AEStronglyMeasurable' m 0 μ

Next tactic:
exact StronglyMeasurable.aeStronglyMeasurable' (@stronglyMeasurable_zero _ _ m _ _)
/-- The integral of the conditional expectation `condexpL1` over an `m`-measurable set is equal to
the integral of `f` on that set. See also `set_integral_condexp`, the similar statement for
`condexp`. -/
theorem set_integral_condexpL1 (hf : Integrable f μ) (hs : MeasurableSet[m] s) :
    ∫ x in s, condexpL1 hm μ f x ∂μ = ∫ x in s, f x ∂μ := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hf : Integrable f
hs : MeasurableSet s
⊢ ∫ (x : α) in s, ↑↑(condexpL1 hm μ f) x ∂μ = ∫ (x : α) in s, f x ∂μ

Next tactic:
simp_rw [condexpL1_eq hf]
/-- The integral of the conditional expectation `condexpL1` over an `m`-measurable set is equal to
the integral of `f` on that set. See also `set_integral_condexp`, the similar statement for
`condexp`. -/
theorem set_integral_condexpL1 (hf : Integrable f μ) (hs : MeasurableSet[m] s) :
    ∫ x in s, condexpL1 hm μ f x ∂μ = ∫ x in s, f x ∂μ := by
  simp_rw [condexpL1_eq hf]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hf : Integrable f
hs : MeasurableSet s
⊢ ∫ (x : α) in s, ↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hf)) x ∂μ = ∫ (x : α) in s, f x ∂μ

Next tactic:
rw [set_integral_condexpL1Clm (hf.toL1 f) hs]
/-- The integral of the conditional expectation `condexpL1` over an `m`-measurable set is equal to
the integral of `f` on that set. See also `set_integral_condexp`, the similar statement for
`condexp`. -/
theorem set_integral_condexpL1 (hf : Integrable f μ) (hs : MeasurableSet[m] s) :
    ∫ x in s, condexpL1 hm μ f x ∂μ = ∫ x in s, f x ∂μ := by
  simp_rw [condexpL1_eq hf]
  rw [set_integral_condexpL1Clm (hf.toL1 f) hs]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hf : Integrable f
hs : MeasurableSet s
⊢ ∫ (x : α) in s, ↑↑(Integrable.toL1 f hf) x ∂μ = ∫ (x : α) in s, f x ∂μ

Next tactic:
exact set_integral_congr_ae (hm s hs) (hf.coeFn_toL1.mono fun x hx _ => hx)
theorem condexpL1_smul (c : 𝕜) (f : α → F') : condexpL1 hm μ (c • f) = c • condexpL1 hm μ f := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
⊢ condexpL1 hm μ (c • f) = c • condexpL1 hm μ f

Next tactic:
refine' setToFun_smul _ _ c f
theorem condexpL1_smul (c : 𝕜) (f : α → F') : condexpL1 hm μ (c • f) = c • condexpL1 hm μ f := by
  refine' setToFun_smul _ _ c f
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f✝ g : α → F'
s : Set α
c : 𝕜
f : α → F'
⊢ ∀ (c : 𝕜) (s : Set α) (x : F'), (condexpInd F' hm μ s) (c • x) = c • (condexpInd F' hm μ s) x

Next tactic:
exact fun c _ x => condexpInd_smul' c x
theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
    (hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hfm : AEStronglyMeasurable' m f μ
hfi : Integrable f
⊢ ↑↑(condexpL1 hm μ f) =ᵐ[μ] f

Next tactic:
rw [condexpL1_eq hfi]
theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
    (hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f := by
  rw [condexpL1_eq hfi]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hfm : AEStronglyMeasurable' m f μ
hfi : Integrable f
⊢ ↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hfi)) =ᵐ[μ] f

Next tactic:
refine' EventuallyEq.trans _ (Integrable.coeFn_toL1 hfi)
theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
    (hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f := by
  rw [condexpL1_eq hfi]
  refine' EventuallyEq.trans _ (Integrable.coeFn_toL1 hfi)
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hfm : AEStronglyMeasurable' m f μ
hfi : Integrable f
⊢ ↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hfi)) =ᵐ[μ] ↑↑(Integrable.toL1 f hfi)

Next tactic:
rw [condexpL1Clm_of_aestronglyMeasurable']
theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
    (hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f := by
  rw [condexpL1_eq hfi]
  refine' EventuallyEq.trans _ (Integrable.coeFn_toL1 hfi)
  rw [condexpL1Clm_of_aestronglyMeasurable']
  

Proof state:
case hfm
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹¹ : IsROrC 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
inst✝⁶ : NormedSpace ℝ F'
inst✝⁵ : CompleteSpace F'
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
f g : α → F'
s : Set α
hfm : AEStronglyMeasurable' m f μ
hfi : Integrable f
⊢ AEStronglyMeasurable' m (↑↑(Integrable.toL1 f hfi)) μ

Next tactic:
exact AEStronglyMeasurable'.congr hfm (Integrable.coeFn_toL1 hfi).symm
theorem condexpL1_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    condexpL1 hm μ f ≤ᵐ[μ] condexpL1 hm μ g := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹⁵ : IsROrC 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : NormedAddCommGroup F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : CompleteSpace 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
hm : m ≤ m0
inst✝⁴ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
E : Type u_8
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
⊢ ↑↑(condexpL1 hm μ f) ≤ᵐ[μ] ↑↑(condexpL1 hm μ g)

Next tactic:
rw [coeFn_le]
theorem condexpL1_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    condexpL1 hm μ f ≤ᵐ[μ] condexpL1 hm μ g := by
  rw [coeFn_le]
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹⁵ : IsROrC 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : NormedAddCommGroup F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : CompleteSpace 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
hm : m ≤ m0
inst✝⁴ : SigmaFinite (Measure.trim μ hm)
f✝ g✝ : α → F'
s : Set α
E : Type u_8
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
⊢ condexpL1 hm μ f ≤ condexpL1 hm μ g

Next tactic:
have h_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x : E, 0 ≤ x → 0 ≤ condexpInd E hm μ s x :=
    fun s hs hμs x hx => condexpInd_nonneg hs hμs.ne x hx