-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
theorem ext_on_measurableSpace_of_generate_finite {α} (m₀ : MeasurableSpace α) {μ ν : Measure α}
    [IsFiniteMeasure μ] (C : Set (Set α)) (hμν : ∀ s ∈ C, μ s = ν s) {m : MeasurableSpace α}
    (h : m ≤ m₀) (hA : m = MeasurableSpace.generateFrom C) (hC : IsPiSystem C)
    (h_univ : μ Set.univ = ν Set.univ) {s : Set α} (hs : MeasurableSet[m] s) : μ s = ν s

Proof state:
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ μ s = ν s

Tactic:
haveI : IsFiniteMeasure ν := by
  constructor
  rw [← h_univ]
  apply IsFiniteMeasure.measure_univ_lt_top

Result:
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
this : IsFiniteMeasure ν
⊢ μ s = ν s

Proof state:
case empty
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
⊢ μ ∅ = ν ∅

Tactic:
simp

Result:
no goals

Proof state:
case measure_univ_lt_top
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ μ univ < ⊤

Tactic:
apply IsFiniteMeasure.measure_univ_lt_top

Result:
no goals

Proof state:
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ IsFiniteMeasure ν

Tactic:
constructor

Result:
case measure_univ_lt_top
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ ν univ < ⊤

Proof state:
case measure_univ_lt_top
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ ν univ < ⊤

Tactic:
rw [← h_univ]

Result:
case measure_univ_lt_top
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
⊢ μ univ < ⊤

Proof state:
case compl
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
t : Set α
htm : MeasurableSet t
iht : μ t = ν t
⊢ μ tᶜ = ν tᶜ

Tactic:
rw [measure_compl (h t htm) (measure_ne_top _ _), measure_compl (h t htm) (measure_ne_top _ _),
  iht, h_univ]

Result:
no goals

Proof state:
case iUnion
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
f : ℕ → Set α
hfd : Pairwise (Disjoint on f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf : ∀ (i : ℕ), μ (f i) = ν (f i)
⊢ μ (⋃ i, f i) = ν (⋃ i, f i)

Tactic:
simp [measure_iUnion, hfd, h _ (hfm _), ihf]

Result:
no goals

Proof state:
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
hs : MeasurableSet s
this : IsFiniteMeasure ν
⊢ μ s = ν s

Tactic:
induction s, hs using induction_on_inter hA hC with
| empty => simp
| basic t ht => exact hμν t ht
| compl t htm iht =>
  rw [measure_compl (h t htm) (measure_ne_top _ _), measure_compl (h t htm) (measure_ne_top _ _),
    iht, h_univ]
| iUnion f hfd hfm ihf =>
  simp [measure_iUnion, hfd, h _ (hfm _), ihf]

Result:
no goals

Proof state:
case basic
α : Type u_5
m₀ : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hμν : ∀ s ∈ C, μ s = ν s
m : MeasurableSpace α
h : m ≤ m₀
hA : m = generateFrom C
hC : IsPiSystem C
h_univ : μ univ = ν univ
s : Set α
this : IsFiniteMeasure ν
t : Set α
ht : t ∈ C
⊢ μ t = ν t

Tactic:
exact hμν t ht

Result:
no goals