-- Mathlib/Analysis/Convex/Integral.lean
theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
    f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
      ∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ

Proof state:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
by_cases h₀' : μ tᶜ = 0

Result:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : μ tᶜ = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Proof state:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
this : ⨍ (x : α), f x ∂μ = ⨍ (x : α) in t, f x ∂μ
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
rw [this, measure_smul_setAverage _ (measure_ne_top μ _)]

Result:
no goals

Proof state:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : μ t = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ENNReal.zero_toReal, zero_smul]

Result:
no goals

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
ht' : μ t < ⊤
⊢ ∫ (x : α) in t, f x ∂μ = ∫ (x : α) in t, const α (⨍ (x : α), f x ∂μ) x ∂μ

Tactic:
clear ht'

Result:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
⊢ ∫ (x : α) in t, f x ∂μ = ∫ (x : α) in t, const α (⨍ (x : α), f x ∂μ) x ∂μ

Proof state:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : t =ᶠ[ae μ] univ
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
rw [restrict_congr_set h₀', restrict_univ, measure_congr h₀', measure_smul_average]

Result:
no goals

Proof state:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
this : ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in t, f x ∂μ) (⨍ (x : α) in tᶜ, f x ∂μ)
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
rw [← H t ht h₀ h₀', openSegment_same, mem_singleton_iff] at this

Result:
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
this : ⨍ (x : α), f x ∂μ = ⨍ (x : α) in t, f x ∂μ
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ¬∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ)

Tactic:
push_neg at H

Result:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ)

Proof state:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
⊢ ∫ (x : α) in t, f x ∂μ = ∫ (x : α) in t, const α (⨍ (x : α), f x ∂μ) x ∂μ

Tactic:
simp only [const_apply, setIntegral_const]

Result:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Proof state:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : μ tᶜ = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ

Tactic:
rw [← ae_eq_univ] at h₀'

Result:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : t =ᶠ[ae μ] univ
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ